## research papers

## Synchrotron radiation loss of a laser accelerator based on an inverse electron cyclotron resonance maser

^{a}IDD Laboratory, School of Physics, Devi Ahilya Vishwavidyalaya, Indore 452001, India^{*}Correspondence e-mail: roma_khullar@yahoo.co.in

A laser accelerator based on an electron cyclotron resonance maser is described. Important losses that give rise to saturation length and saturation power are included in the analysis. The results are compared with results from an inverse free-electron laser accelerator.

### 1. Introduction

The field strength in a typical RF accelerating cavity is considerably smaller than that which can be achieved in modern advanced lasers. This leads to growing interest in laser accelerators that produce charged particle beams at highly relativistic energies using intense coherent electromagnetic waves. In the past, various mechanisms and accelerator schemes have been proposed and adopted for laser accelerators. Successful experiments on free-electron lasers have seen the development of inverse free-electron laser (IFEL) accelerators (Palmer, 1972; Courant *et al.*, 1985; Musumeci *et al.*, 2005; Duris *et al.*, 2012, 2014; Khullar *et al.*, 2015), achieving from laser beams to electrons. In the IFEL scheme, the relativistic electrons move through an undulator magnet collinearly with an electromagnetic wave. The undulator magnet produces a small transverse velocity in the direction parallel to the electric-field vector of the collinear propagating electromagnetic wave, so that between the electron and the electromagnetic wave is possible. The undulator was tapered using an IFEL scheme in order to accelerate electrons to energies of the order of GeV (Sprangle & Tang, 1981; Pellegrini, 1982). An alternative scheme has been developed that involves the laser acceleration of electrons in a uniform magnetic field based on the electron cyclotron resonance maser principle (Colson & Ride, 1979; McDermott *et al.*, 1985; Shpitalnik *et al.*, 1991, 1992; Chen, 1991, 1992; Sprangle *et al.*, 1983; Lie, 2002; Mirzanejhad *et al.*, 2006). In this scheme, an electron with an initial perpendicular velocity component is guided by the uniform axial field. Initially, the electrons undergo cyclotron motion and they are uniformly arranged in a rotational phase on the circle. The Larmar radius is defined as , where is the initial perpendicular velocity of the electron and Ω_{c} = *eB*/*m*γ is the relativistic cyclotron frequency. After interaction with the electromagnetic fields, some electrons that are decelerated in the electric field rotate faster to accumulate phase lead. Some electrons that are accelerated rotate slowly to accumulate phase lag. This phase bunching leads to a coherent radiation mechanism of the electron cyclotron resonance maser. The is proportional to that produced, , where is the electron transverse velocity component and is the transverse electric field.

In this paper, we consider laser acceleration of electrons based on the inverse electron cyclotron resonance maser principle. We consider cyclotron radiation loss as an important factor for the net acceleration of the laser accelerator and we include this in our analysis. We solve relativistic Lorentz force equations in a uniform axial magnetic field, assuming circularly polarized electromagnetic waves of constant amplitude. An analytical solution has been obtained for the maximum energy attained in the laser accelerator. The solutions yield a dependence of the electron perpendicular velocity component on the maximum energy attainable, the saturation length and the saturation energy of the laser accelerator. It has been shown that in a magnetic field of 16 T, power of 400 GW, a 10.6 µm wavelength laser with an optimum electron beam with a Larmar radius in the range 10–30 µm can accelerate electrons to ∼20 GeV in less than 5 m.

### 2. Accelerator equations

The particle motion and energy change of a single charged particle in the presence of a uniform axial magnetic field and a laser are governed by relativistic Lorentz force equations. We consider a uniform magnetic field produced by a solenoid and given by

An electromagnetic wave of a circularly polarized laser is described by

Here, *k*_{r} and ω_{r} denote the wavenumber and the frequency of the electromagnetic wave, respectively. The equation describing the motion of the electrons in the IFEL can be derived from the Lorentz equation of motion,

In component form, equation (2) reads

where and Equation (3) for the electron velocity is read as

and the solutions are

where

Substituting β_{x} and β_{y} from equations (5*a*) and (5*b*) into the relation β^{2} = , we obtain the longitudinal velocity,

where ψ = *k*_{r}*z* − ω_{r}*t* + Ω_{c}*t*. The longitudinal coordinate can be evaluated by integrating the above expression to obtain

The change in electron energy is given by

Using equation (5), the change in electron energy is given by

where *A* = *eE*_{0}/*m _{e}c*

^{2}and the phase ψ =

*k*

_{r}

*z*− ω

_{r}

*t*+ Ω

_{c}

*t*. The resonant condition for dψ/d

*z*= 0 is

In an accelerator, the effects of synchrotron-radiation loss are important and give the saturation length and saturation power of the device,

where

and = dβ/d*t*. Using equation (5), the synchrotron-radiation loss of the electron is given by

where *r*_{e} = *e*^{2}/*m*_{e}*c*^{2} is the classical electron radius. We can write the accelerator equation, including the synchrotron loss term, as

Using the resonant condition in equation (14), Ω_{c} ≃ ω_{r}/2γ^{2}. The accelerator equation becomes

Using equation (15) for the undulator length, we obtain

The maximum energy attainable from the device () occurs when we set dγ/d*z* = 0. This gives

where = *A*sinψ. Equation (16), in terms of the maximum attainable energy, can be written as

Using the standard integral results,

the solution for equation (18) is written as

where γ_{0} is the initial energy of the electron.

### 3. Results and discussion

A laser accelerator based on inverse electron cyclotron resonance maser is described. The effects of synchrotron radiation losses are included in the analysis. The relativistic Lorentz force equation is solved analytically for an electron beam having an initial finite perpendicular velocity. The electron beam travels in a helical path in the axial magnetic field and exchanges energy with a copropagating circularly polarized laser field. The electric field of the laser can be calculated using

where *W* is the laser power and *w*_{0} is the laser spot size. A 400 GW, 10.6 µm-wavelength laser focused to a 240 µm spot size gives an of *E* = 1.33 × 10^{6} statV cm^{−1} [1 V = (1/3) × 10^{−2} statV]. The maximum energy attainable from the laser accelerator can be estimated from equation (17). Considering *r*_{e} = 2.8179 × 10^{−13} cm, *A* = 771 cm^{−1} and sinψ = 0.866, we rewrite equation (17) as

For and the maximum attainable energy from the accelerator is = 23 GeV (Fig. 1).

The maximum attainable energy in the laser accelerator is inversely proportional to the square root of the perpendicular velocity, *i.e.* . An electron beam with a lower can provide high energy but with a lower gradient [equation (17)]. This implies that a lower value of would require a large saturation length and is therefore undesirable. A higher value of implies efficient interaction with the laser and the saturation length is smaller and can result in a compact laser accelerator device. The energy gain curves are drawn in Fig. 2 for γ_{0} = 30 and for a value in the range = 0.1–0.3. Higher values reach saturation quickly but the saturation energy is substantially lower. The saturation length and saturation energy are plotted in Figs. 3 and 4, respectively; for , the values are 5 m and ∼21 GeV. To estimate the analytical expression for the acceleration gradient, we consider equation (16) without synchrotron-radiation losses, *i.e.*

For *mc*^{2} = 0.511 MeV, *A* = 771 cm^{−1} and sinψ = 0.866, we rewrite the equation in practical units as

For , the accelerating gradient is 6.8 GeV m^{−1}. Using the resonant condition from equation (10) (*i.e.* Ω_{c} = ω_{r}/2γ^{2}), the magnetic field in our calculation is 16.8 T for γ_{0} = 30. In Fig. 5 we calculated the accelerating gradient. A linear fit for the calculation yields

For = 0.2, the calculation gives 3.93 GeV m^{−1}, whereas the fit predicts a value of 3.91 GeV m^{−1}. The discrepancy between the calculation and the fit formula is almost 2%. The Larmar radius of the electron at this value can be calculated from

For in the range 0.1–0.3, the Larmar radius will be in the range 10–30 µm.

### Funding information

This work has been carried out with financial support from SERB, Government of India (grant No. EMR/2014/00120).

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