2.1 Model parameters

Let us next discuss the basic model parameters. (Unfortunately, there is some confusion in standard definitions used in literature.)

The model starts with an assumption that guiding magnetic field lines are perturbed by a packet of linearly polarized Alfvén waves of intensity $B_w$ and frequency $\omega$. In highly magnetized force-free plasma Alfvén waves propagating along the magnetic field are nearly luminal, $v_A \approx c$.

The first parameter is dimensionless wave intensity. We chose notation $a_0$, which is standard in the laser community and sometimes used in the FEL community as well,

(2.1)\begin{equation} a_0 = \frac{e B_w}{m_e c \omega} \gg 1 \end{equation}

It should be remarked that in the FEL literature parameters $K$ and $a_0$ are often used interchangeably. In this paper we follow the nomenclature used for $K$ in (2.4) below in Jackson (Reference Jackson1975, paragraph 14.7)

For the guide-field dominated regime, another parameter is the relative intensity

(2.2)\begin{equation} \delta= \frac{B_w}{B_0}\ll 1, \end{equation}

where $B_0$ is a guide field. We assume that the wave is relatively weak, $\delta \ll 1$.

A (reconnection-generated) beam of particles with Lorentz factor $\gamma _b$ propagates along the rippled magnetic field in a direction opposite to the direction of the Alfvén waves. In the frame of the beam the waves are seen with $k_{w}'= 2 \gamma _b k_{w}$. In the guide-field dominated regime the cyclotron frequency associated with the guide field is much larger than the frequency of the wave in the beam frame, and the cyclotron frequency associated with the fluctuating field; hence,

(2.3)\begin{equation} \varOmega_0 \gg k_{w}' c,\varOmega_w, \end{equation}

where $\varOmega _0 = e B_0/(m_e c)$ is the cyclotron frequency (non-relativistic) of the guide field and $\varOmega _w = e B_w/(m_e c)$ is the cyclotron frequency associated with the wiggler field.

Another important parameter, defined by Jackson (Reference Jackson1975, paragraph 14.7), is the wiggler-undulator parameter

(2.4)\begin{equation} K = \delta \gamma_b. \end{equation}

This parameter is related to the magnitude of the wiggler-induced oscillations in the beam trajectory, which is also related to the opening angle of the cone of the generated radiation. When $K \gg 1$, this oscillation is large and the pump field is sometimes referred to as a ‘wiggler’. In the opposite regime where $K \ll 1$ the pump field is sometimes referred to as an ‘undulator’. In this paper we will refer to the pump field as a wiggler throughout.

The $K$ parameter (2.4) is a product of two quantities, relative amplitude $\delta \ll 1$ and Lorentz factor $\gamma _ b \gg 1$, so generally its values can be either large or small. A relativistically moving electron emits in a cone with opening angle $\Delta \theta \sim 1/\gamma$. In the $K \ll 1$ regime that opening angle is much larger than the variation in the bulk direction of emission at different points in the trajectory; see figure 1, left panel. The radiation detected by an observer is an almost coherent superposition of the contributions from all the oscillations in the trajectory at a frequency

(2.5)\begin{equation} \omega_C =\gamma_b^2 ( c k_{w}) \times \left\{\begin{array}{@{}cc} 4 & ({\rm EM\ wiggler})\\ 2 & ({\rm static\ wiggler}) \end{array}\right. \end{equation}

We note that the axial velocity in the strong axial guide-field regime is nearly constant and close to the speed of light for high energy electrons; see figure 7 in § 4.3. Hence, the resonant frequency is also nearly constant.

Figure 1.

Different regimes of wave–particle interaction depending on the parameter $K$ (2.4). Grey cones illustrate the instantaneous emission cone.

In the $K\gg 1$ regime the variations in the direction of emission are much larger than the angular width of the emission at any point in the trajectory; see figure 1(b). As a result, an observer located within angle $\leq \delta$ with respect to the overall guide field see periodic bursts of emission with typical frequency

(2.6)\begin{equation} \omega_c \approx \gamma_b^2K \times ( c k_{w}). \end{equation}

Since $K \gg 1$ in this regime, the resulting frequency is higher than that for the case where $K \ll 1$. In this paper we work in the $K \ll 1$ regime, which is the regime more typical of FELs.

Another terminology issue: here we contrast the term ‘Compton’ with ‘curvature’, not with ‘Raman’ regime, which is Compton-like scattering, but on collective plasma fluctuations.

In this paper we work in the regime $K \ll 1$ (this is the usual regime of FELs). The scattered frequency is then given by (2.5); below we drop the subscript $C$.

2.2 Overview of main results of Lyutikov (Reference Lyutikov2021)

Lyutikov (Reference Lyutikov2021) developed a model of the generation of coherent radio emission in the Crab pulsar, magnetars and FRBs whereby the emission is produced by a reconnection-generated beam of particles via a variant of the FEL mechanism, operating in a weakly turbulent, guide-field dominated plasma. The guide-field dominance is a key new feature that distinguishes this regime from the conventional laboratory FELs: a (reconnection-generated) beam of particles with Lorentz factor $\gamma _b$ propagates along the wiggled magnetic field in a direction opposite to the direction of the Alfvén waves. In the frame of the beam the waves are seen with $k_{w,b}= 2 \gamma _b k_{w}$. Guide-field dominance requires $2 \gamma _b k_{w} c \ll \varOmega _0$.

The key results of Lyutikov (Reference Lyutikov2021) are as follows.Footnote ¹

1. (i) The interaction Hamiltonian (this is particularly important for the present work, as it describes evolution of the instability and its saturation). Particle motion in the combined fields of the wiggler $B_w$, the EM wave $E_{EM}$ (both with wave vector $k_{w}'$ in the beam frame) and the guide field $B_0 \gg E_w, E_{EM}$ can be described by a simple ponderomotive Hamiltonian (Lyutikov Reference Lyutikov2021, (39))

   (2.7)\begin{equation} {\mathcal{H}} = \frac{\beta_z^2}{2} + \delta \left(\frac{ E_{EM}}{B_0} \right) \left( \frac{\varOmega_0}{ k_w c} \right) \sin ^2( k_{w}' z), \end{equation}
   where $\beta _z$ is axial velocity; see figure 2. The second term is the ponderomotive potential. The Hamiltonian formulation allows powerful analytical methods to be applied to the system (adiabatic invariant, phase space separatrix etc). This is especially important for the estimates of the nonlinear saturation, one of the main goals of the present work.

2. (ii) The growth rate of the parametric instability is (Lyutikov Reference Lyutikov2021, (62))

   (2.8)\begin{equation} \varGamma = \left( \left( \frac{ E_w}{B_0} \right) \left( \frac{ E_{EM}}{B_0} \right) {\varOmega_0}{ k_w c} \right)^{1/2} \propto B_0^{{-}1/2}. \end{equation}
   Importantly, it is only mildly suppressed by the strong guide field.

3. (iii) Saturation level. The ponderomotive potential increases linearly with EM wave intensity, while energy density of EM waves increases quadratically. The corresponding saturated velocity jitter (Lyutikov Reference Lyutikov2021, (73))

   (2.9)\begin{equation} \beta_{S} = \frac{\delta}{ \sqrt{2 \gamma}} \frac{\omega_p}{k_w c} \end{equation}

4. (iv) The guide-field dominance plays a dual role. First, it reduces the growth rate, but only mildly, ${\propto } B_0^{-1/2}$ (2.8). What is more important, the strong guide field helps to maintain beam coherence. Without the guide field, particles with different energies follow different trajectories and quickly lose coherence even for a small initial velocity spread. In contrast, in the guide-field dominated regime all particles follow, basically, the same trajectory. Hence, coherence is maintained as long as the velocity spread in the beam frame is $\Delta \beta \leq 1$ (see figure 13 of Lyutikov Reference Lyutikov2021). As a result, the model requires only a mildly narrow distribution of the beam's particles, $\Delta p /p_0 \leq 1$ and the spectrum of turbulence $\Delta k_{w,b}/k_{w,b} \leq 1$

5. (v) The model operates in a very broad range of the neutron star's parameters: the model is independent of the value of the magnetic field. It is thus applicable to a broad variety of NSs, from fast spin/weak magnetic field millisecond pulsars to a slow spin/supercritical magnetic field in magnetars, and from regions near the surface up to (and a bit beyond of) the light cylinder.

Figure 2.

Particle dynamics for FEL in the guide-field dominated plasma with the ponderomotive potential (2.7) (arbitrary unites chosen to illustrate the trajectory Lyutikov Reference Lyutikov2021). (a) Three-dimensional rendering of particle trajectories in the beam frame (a particle experiences a saddle-like trajectory). (b) Trajectory of trapped particles in phase $\beta _z$–$z$ plane; velocity is normalized to the separatrix $\beta _{S}$ (2.9).

3.1 The codes

In this work we performed simulations of the interaction of a single-charged relativistic beam with the wiggler using the FEL code ONEDFEL (Freund & Antonsen Reference Freund and Antonsen2024). ONEDFEL is a time-dependent code that simulates the FEL interaction in one dimension. The radiation fields are tracked by integration of the wave equation under the slowly varying envelope approximation. As such, the wave equation is averaged over the fast time scale under the assumption that the wave amplitudes vary slowly over a wave period. The dynamical equations are a system of ordinary differential equations for the mode amplitudes of the field and the Lorentz force equations for the electrons that are integrated simultaneously using a fourth order Runge–Kutta algorithm. Time dependence is treated by including multiple temporal ‘slices’ in the simulation that are separated by an integer number of wavelengths. The numerical procedure is that each slice is advanced from $z \to z + \Delta z$ separately by means of the Runge–Kutta algorithm. Time dependence is imposed as an additional operation by using the forward time derivative as an additional source term to treat the slippage of the radiation field with respect to the electrons. Slippage occurs at the rate of one wavelength per undulator period in the low gain regime and after saturation of the high gain regime but at the rate of one third of a wavelength per undulator period in the exponential gain regime (Bonifacio et al. Reference Bonifacio, Casagrande, Cerchioni, Salvo Souza, Pierini and Piovella1990; Saldin, Schneidmiller & Yurkov Reference Saldin, Schneidmiller and Yurkov1995; Freund & Antonsen Reference Freund and Antonsen2024). ONEDFEL self-consistently describes slippage in each of these regimes. The simplest way to accomplish this is to use a linear interpolation algorithm to advance the field from the $(i -1)$th slice to the $i$th slice. Using this procedure ONEDFEL can treat electron beams and radiation fields with arbitrary temporal profiles and it is possible to simulate complex spectral properties.

Simulations, effectively, work in the lab frame, The general set-up consists of the following.

1. (i) Guide magnetic field $B_0$ (as strong as numerically possible).

2. (ii) A wiggler with wavenumber $k_w$ and relative amplitude $\delta = B_w/B_0 \ll 1$ is as EM wave (with adiabatic turning on); wiggler's frequency in the beam frame is below the cyclotron frequency associated with the guide field, $\omega _w \sim \gamma _b \ll \omega _{B_0}$ (but can be comparable to the cyclotron frequency of the wiggler, $\omega _{B_w}$).

3. (iii) A charged beam with a ‘solid’ (dead) neutralizing background; the corresponding Alfvén wave is relativistic, $v_A \sim c$. The beam is initially propagating along the magnetic field (no gyration).

4. (iv) The pulse duration is much longer than the wiggler wavelength.

By using a pure EM wave, and not as a self-consistent Alfvén wave, eliminates complications related to setting the correct particle currents. In the highly magnetized regime $\sigma \gg 1$, Alfvén waves are nearly luminal (here $\sigma = B^2/(4{\rm \pi} \rho c^2)$ is the magnetization parameter (Kennel & Coroniti Reference Kennel and Coroniti1984)).

Let us next comment on the applicability of the one-dimensional regime. The post-eruption magnetic field lines are mostly radial. Development of a coronal mass ejection (CME), accompanying magnetospheric FRBs, leads to opening of the magnetosphere from radii $R_0$, much smaller than the light cylinder (Sharma, Barkov & Lyutikov Reference Sharma, Barkov and Lyutikov2023); see figure 3. After the generation of a CME the magnetosphere becomes open, with nearly radial magnetic field lines for $r\geq R_O$

Figure 3.

Post-flare opening of the magnetosphere (Sharma et al. Reference Sharma, Barkov and Lyutikov2023). Colour represent values of $r \sin \theta B_{\phi }$ and lines are poloidal field lines. The spin parameter is $\varOmega =0.2$, so that before the ejection of a CME the light cylinder is at $R_{LC} =5$ (the dashed white line is the last closed magnetic field). Post-flare magnetosphere is open, starting $R_O \ll R_{LC}$. At $r \geq R_O$ the magnetosphere has a monopolar-like magnetic field structure. The appearance of the plasmoids (‘ejected plasmoid’) is not important for the present work.