Spectral Bandwidth Reduction of Thomson Scattered Light by Pulse Chirping

Based on single particle tracking in the framework of classical Thomson scattering with incoherent superposition, we developed a fully relativistic, three dimensional numerical code that calculates and quantifies the characteristics of emitted radiation when a relativistic electron beam collides head-on with a focused counter-propagating intense laser field. The developed code has been benchmarked against analytical expressions, based on the plane wave approximation to the laser field, derived in (1). For sufficiently long duration laser pulses, we find that the scattered radiation spectrum is broadened due to interferences arising from the pulsed nature of the laser. We show that by appropriately chirping the scattering laser pulse, the spectral broadening could be minimized.


Introduction
Intense, tunable, ultra-short, collimated, polarized, and quasi-mono-energetic radiation in the xray and gamma-ray region of the electromagnetic spectrum has potential applications in broad disciplines that extends to, but is not limited to, natural sciences and health sciences. Such sources, which may be referred to as laser synchrotron sources (LSS), may be realized with the inverse Compton scattering, or Thomson scattering, process (2)(3)(4). In inverse-Compton scattering, an intense laser field with frequency 0  is scattered by a counter-propagating high energy ( e E ) electron beam. In the limit where sources (5), which are typically un-polarized, with broad energy spectrum, LSS sources have a high degree of polarization (6), which is determined by the polarization of the scattering laser pulse, with relatively narrow band spectrum.
The generation of bright picosecond duration x-rays and gamma-rays through inverse-Compton scattering has already been demonstrated with head-on collision of intense laser pulses synchronized to picosecond-duration, high-energy electron beams generated with conventional radio frequency (RF) based accelerators (7,8). Ultra-short x-rays, with moderate brightness, have also been demonstrated from the inverse-Compton scattering of femto-second lasers in a 90 0 scattering geometry with a synchronized, RF-accelerated, high-energy electron beam (3). In addition to the experimental demonstration of x-ray and gamma-ray sources, the backscattered radiation was used as a diagnostic tool for the electron beam (9,10) to generate polarized positrons (11) from dense targets, and to demonstrate nuclear fluorescence (proving the usefulness of the source for discerning isotope specific elements) (12,13).
The theory of Thomson backscattering of an infinitely long electromagnetic field by a relativistic electron beam has been well documented (14)(15)(16). Moreover, extensive theoretical simulations have also been performed for scattering from a pulsed plane wave electromagnetic field. We extend these previous works by including the realistic 6-dimensional nature of the electron beam as well as the three-dimensional nature of the focused electromagnetic pulse with curved wave fronts. This treatment includes spectral broadening due to wave front curvature and finite temporal duration of the scattering laser and broadening associated with the transverse and longitudinal emittances of the electron beam. Such a detailed calculation of Thomson scattering is necessary to provide a framework for experimental events and to guide the design of Thomson-scattered x-ray sources.
In this paper, we discuss a numerical code that calculates the scattered radiation during the interaction of an intense laser pulse with an electron beam. In addition to benchmarking the code against previously reported results, we use it to demonstrate a technique to reduce the spectral bandwidth of Thomson scattered light by means of chirping the incident scattering laser pulse.
The paper is organized as follows: Section 1 discusses the core ingredients of the developed code; in Section 2, a comparison of the numerically calculated and analytically obtained radiation energy is made; Section 3 discusses the effect of finite temporal width of the laser on the scattered spectrum as well as a method to overcome broadening due to the pulsed nature of the scattering laser. Summarized results are presented in Section 4.

Modeling
The three-dimensional and fully relativistic Thomson code is divided as follows: (1)

Relativistic Electron Dynamics
Once the electron beam is described accurately by a sampled 6-dimensional phase space distribution, the dynamics of each electron in the laser field is calculated by solving the relativistic equations of motion given by, m is the rest mass of an electron, q the charge of an electron, p the particle momentum, laser E and laser B are the laser magnetic and electric field vectors, and  is the relativistic Lorentz factor.
The code we developed can calculate the laser field to a high degree of accuracy, i.e., nonparaxial field terms, relevant to a fast focus; terms up to order 7 are included. For simplification, the results presented in this paper are calculated with a plane wave laser field; allowing direct comparison with analytic solutions.
We use a 4 th -order Runge-Kutta ordinary differential equation solver with relative errortolerance threshold of 10 −6 , a local error threshold of 10 −12 , and a time step typically of the order 10 −4 femtosecond. Since the electron beams considered in this paper are relativistic and have low density 3 16 3 ( / 10 cm ) e n    space charge forces are neglected (1,17). In the absence of the laser field, the electron beam trajectory is assumed to be ballistic. Radiation damping is also not accounted for in these calculations, since the energy radiated per cycle by an electron is small compared to the energy of the electron.

Radiation Calculation
Once the dynamics of each electron are obtained, the energy density radiated per unit frequency  and solid angle  by a single electron moving in the intense laser field can be described by the classical formula (18) To benchmark our code, we used the general analytic expressions of the radiated energy density, derived by integrating the Lienard-Wiechert potentials (14,19). Of particular interest is the limit of low strength laser fields, 0 1 a  , where the radiation is dominated by the first harmonic. For small observation angle  and in the limit of non-relativistic scattering laser intensity ( 0 1 a  ), the energy radiated by a single electron per unit solid angle  and per unit frequency  may be written as (20, 21): z  -axis is backscattered by the electron beam, and the resulting backscattered energy density is shown in Figure 1 (b). The radiation calculated numerically with the developed code is in close agreement with the one obtained analytically with equation (2.1), see Figure 1  , and the corresponding Thomson bandwidth is 55.5%.

Scattering from a Pulsed Laser and Spectral Broadening
Previous work investigated the effect of beam shapes on the Thomson scattered spectrum (22)(23)(24). In particular, the pulsed nature of the laser pulse has been shown to introduce spectral substructures within the radiated harmonics. When an electron interacts with a pulsed laser pulse, The spectral difference in the radiation from the increasing and decreasing parts of the laser pulse results in spectral interference of the radiated field, creating oscillations in the radiated spectrum (24). In a recent study (22), it is shown that the number of oscillations is proportional to the laser intensity and temporal duration, i.e.,  This non-linear oscillation in the spectrum might be minimized by using an appropriately chirped laser pulse. We propose a chirped laser pulse whose frequency changes with time as 2 00 ( ) 2 / 3 (1 ( ( ) / ) / 2) t a t a   . This would ensure that radiated frequencies during the period of the laser pulse will be identical, and the spectral oscillations resulting from the spectral interference of the different radiated frequencies could be minimized. Since most of the radiation occurs near the peak of the laser intensity, the proposed chirp may be realized with a phase . For the chirped case, the radiated spectrum is dominated by the fundamental with only two less prominent substructures. Such a scheme might be used in the design of narrow band gamma-ray sources from the scattering relativistic electron beams with very small energy spreads, such as electron beams obtained with conventional accelerators.

Conclusion
In summary, based on the single particle trajectory tracking, a fully relativistic 3-dimensional non-linear Thomson scattering code has been developed and benchmarked against analytical expressions for scattering in a plane wave laser field. With a simple switch implemented in the code, the developed code is able to calculate the laser field to a high degree of accuracy.
Furthermore, the code is used to investigate the beam shape effects on the scattered radiated energy. It was found that substructures in the emitted radiation spectrum (due to the pulsed nature of the laser pulse), which have a detrimental effect on the quality of the radiated spectrum, could be minimized by chirping the laser pulse appropriately.