High-brilliance synchrotron radiation induced by the plasma magnetostatic mode

Using multi-dimensional PIC simulations we show that the magnetic undulator-type field of the plasma magnetostatic mode is indeed produced by the interaction of a laser pulse with a relativistic ionization front, as predicted by linear theory for a cold plasma. When the front with this magnetostatic mode is followed by a relativistic electron beam, the interaction of the beam with this magnetic field, produces FEL-type synchrotron radiation, providing a direct signature of the magnetostatic mode. The possibility of generating readily detectable ultrashort wavelength radiation using this mode, by employing state-of-the-art laser systems, is demonstrated, thus opening the way towards experimental observation of the hitherto unseen magnetostatic mode and the use of this plasma FEL mechanism to provide a source of high-brilliance ultrashort wavelength radiation.

The advent of high power lasers has allowed for the exploration of the extraordinary ability of plasmas to sustain strong electric [1] and magnetic [2] fields. Electrostatic plasma waves [3], driven by intense laser pulses have well-defined electric structures that can be used to accelerate electron beams up to the GeV energy [4][5][6][7]. Laser-plasma interactions are also known to provide ultra-strong ( > ∼ 100 T) magnetic fields [2,8]. However, effective methods for generating controlled periodic magnetostatic structures in plasmas are yet to be demonstrated.
In a recent study, it was shown that an overdense ionization front moving nearly at c can frequency upshift microwave radiation confined in a cavity [9]. This study, and earlier theoretical work [10], has led to the understanding of how electromagnetic boundary conditions should be handled at a plane dielectric discontinuity moving normal to the plane. A zero-frequency, zero-electric-field, purely Ampère-law magnetostatic mode (here called the MS mode) was recognized as being important. This mode has hitherto not been seen and is ordinarily very difficult to excite. With a plasma front boundary which is moving at a normal velocity v f , the general phase continuity condition is that the quantity (ω − k · v f ) does not change when changing from one medium to the other in ω − k space [10]. With a fast-moving front and an incident EM (ElectroMagnetic) wave whose frequency ω i is well below the plasma frequency (ω p ) behind the front, the stationary MS mode becomes easy to set up, and the MS mode (which has the same magnetic polarization as the incident EM wave) becomes the dominant repository for the incident energy [10]. (The value of ω p is usually that obtained after rapid and complete ionization by a high frequency (ω 2 d ω 2 p ) ionization front driver laser pulse of the gas behind the ionizing pulse.) The MS mode will have a normal wavevector k M S of (1 + v f /c) times the incident wavevector (in the case of interest this factor is very nearly 2). Since the MS mode does not propagate and cannot exist outside the plasma, it has not yet been detected directly.
For a plane polarized incident EM wave the MS mode in the plasma has the magnetic field geometry of standard magnetic undulators, such as those used for FELs (Free Electron Lasers), and from this arises the key concept of this letter in addressing the MS mode detection problem. One begins by creating a usefully strong MS undulator-type B-field using the sub-plasma-frequency laser wave incident on the counter-propagating ionization front created by a short ionization-front laser drive pulse incident from the opposite side of the plasma. The key new component here is to supply also a highly relativistic electron beam pulse to pass through the stationary MS undulator magnetic field almost immediately after it is produced. This is done in order to produce a pulse of synchrotron radiation more or less in the electron beam direction, thereby confirming the existence of the MS mode. This MS synchrotron radiation will provide both a direct diagnostic of the MS spatial period (from the frequency spectrum peak) and a reasonable estimate of its strength (from the magnitude of the synchrotron signal). Going beyond this basic verification of the MS mode physics brings in the complementary notion of using the concept to produce a powerful undulator magnetic field of much shorter spatial period than is possible to obtain with conventional magnets. Applications might range from condensed matter physics [11,12] to novel radiations sources [13]. Showing how this would work in practice is the purpose of the rest of this paper, as exemplified by the simulation results and the further detailed discussions.
The efficient generation of the MS mode requires a sharp ionization front, i.e. L g k i 1, where L g is the gradient scale of the ionization front and k i is the wavenumber of the incident light [10]. Steep ionization fronts can be generated by the propagation of a short intense laser pulse in a gas jet via tunneling ionization. The relativistic factor of a laser-driven ionization front, in the linear regime, is given by where λ i = 2πc/ω i is the wavelength of the incident radiation and I i its intensity, which is only limited by the threshold for ionization of the background gas. For Hydrogen, for instance, the threshold intensity for tunnel ionization [14] is ∼ 10 13 W/cm 2 for mid-infrared light, allowing for the generation of magnetic-field amplitudes on the order of 100 T.
Present state-of-the-art CO 2 laser systems [15] can deliver pulses with energy i = 170 J and λ i = 10. As the beam electrons wiggle in the undulator structure they emit radiation with wave- , where γ b is the relativistic factor of the beam electrons. The short wavelength (λ u ∼ 5 µm) and high amplitude (B u ∼ 100 T) laser driven MS structure allows for the generation of radiation with features that can provide an evidence of the MS mode (energy determined by the properties of the e-beam and MS structure, narrow spectrum, and high brilliance). The synchrotron radiation emitted by the e-beam traversing the MS mode, and therefore the MS mode itself, can be unambiguously detected through a straightforward null test whereby the e-beam/incident laser pulse are present or not in the experimental set-up. The average radiated energy is P r τ p /2, where τ p is the propagation time of the e-beam in the MS structure. Since the residence time of the e-beam in the MS structure is half the duration of the incident wave, τ i , the total radiated energy can be written as The energy of the radiated photons is which depends only on the energy of the e-beam and on the MS wavelength. We note that while the MS magnetic field can be strong, K is still small due to the short wavelenght of the MS undulator, and, therefore, the photon energy is well approximated by 4γ 2 b times the incident laser photon energy. The total number of photons, N γ = r / γ , is given by A proof-of-principle experiment, demonstrating and measuring the MS mode can be performed using the widely available 1 J class Ti:Sapphire laser systems, whose laser pulses can efficiently generate both the relativistic ionization front and LWFA e-beams with b = 50 MeV and Q b = 30 pC [5]. Using an improved two-stage Raman shifting scheme with > 0.2% efficiency [18], a fraction of the laser energy (0.2 J) can be used to produce pulses with τ i = 4 ps and λ i = 5 µm, which then collide with the ionization front at incident intensities I i N γ 10 5 , peaked at γ 10 keV, which would allow for a clear evidence of the MS mode.
Since the e-beam probing the MS mode propagates in a plasma, it is important to assess the possible beam quality degradation, in terms of energy spread and emittance, due to beam-plasma and laser-plasma instabilities. The relevant beam-plasma instabilities are the two-stream instability (longitudinal) and the Weibel instability (transverse). The two-stream instability can be avoided using short e-beams, [19], where n b is the electron density of the beam. The Weibel instability is strongly suppressed for e-beams narrower than the plasma wavelength λ p = 2πc/ω p [20], or for wider low-density e-beams with n b /n e < 2×10 −3 ( b [MeV]/100)(θ[µrad]/10) 2 [21].
In what concerns laser-plasma instabilities, laser filamentation is avoided for laser powers smaller than the critical power for self-focusing [22], and stimulated Raman scattering [23] will not be driven for ultrashort laser pulses, which are required in order to generate a steep ionization front. Laser wakefields [4,24], which can also have deleterious effects on the e-beam quality, are negligible for a laser normalized vector potential a 0,d 1, since δn e /n e0 1.
Ionization front irregularities can also contribute to a degradation of the MS mode and consequently of the emitted radiation. Erosion of the head of the laser pulse can broaden the initially sharp ionization front and, therefore, the laser should contain enough energy to generate a stable ionization front during the interaction process. The energy depletion length, L dpl , due to ionization can be roughly estimated by equating the laser energy to the ionization energy of a single atom, W ion , times the number of atoms ionized, n 0 σ d τ d  the radiation spectrum exhibits a narrow energy spread ∆ γ / γ = 0.012 (shown in the inset).