# Capabilities of Terahertz Cyclotron and Undulator Radiation from Short Ultrarelativistic Electron Bunches

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## Abstract

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## 1. Introduction

## 2. 1D Model of a Perfect CSR Source

_{z}. Here, e, m, and $\gamma $ are the electron’s charge, mass, and Lorentz factor, respectively, and c is the speed of light. In the second case, we consider the case where the particles also move along helical trajectories, but oscillate with the frequency of forced undulator oscillations ${\omega}_{u}=2\pi {v}_{z}/d$ and cyclotron oscillations are not excited: d is the undulator period and ${v}_{z}$ is the constant longitudinal velocity. In the second case, the transverse particle velocity normalized to the speed of light is determined by the expression (see e.g. [16] and Equations (10) and (11))

## 3. Cyclotron Radiation of an Electron Bunch in a Cylindrical Waveguide

_{b}much smaller than the waveguide radius R

_{w}and an axis coinciding with the axis of the waveguide. At the initial time, the particle density distribution in the longitudinal and transverse directions are described by the Gaussian dependences, and in the simplest case the transverse and longitudinal velocities of all particles are considered equal (later, effects associated with the variation of the particle parameters will be considered). In the study of the dynamics of bunches and radiation in this paper, two computational approaches are used. In the approximate method [13,18], the bunch is represented by a set of solid uniformly charged disks whose centers are initially placed on the waveguide axis; the disks interact with the radiation field and between each other, and the radiation field is represented by the lowest TE

_{11}wave with a smoothly varying amplitude. In a more general and precise spatial-frequency approach WB3D [19], macroparticles are used, which are thrown at the initial moment into the waveguide and interact with each other and with the radiation field represented by a set of waveguide modes excited at any cyclotron harmonic. The interaction of the particles with each other (which is responsible for a correct description of space-charge forces affecting the bunch) is described by linearized formulas, following from the Lienard–Wichert potentials (see [15] for more details).

_{11}mode of a circular waveguide at the fundamental cyclotron harmonic show a good agreement with the results for the plane in free space. Namely, the current frequency and efficiency of radiation for such a disk in the regimes of group synchronism and high-frequency intersection of dispersion characteristics little differ from the corresponding values for radiation of the plane when both radii of the waveguide, R

_{w}, and the disk, R

_{d}, are sufficiently large and the ratio of surface densities of the disk and plane (form-factor) is $\frac{{\sigma}_{d}}{\sigma}=0.8\left(\frac{{R}_{w}^{2}}{{R}_{d}^{2}}\right)$, where the coefficient 0.4 is equal to the norm of the wave. It should be mentioned that according to simulations, the model of a solid disk set [13,18] is a fairly accurate approximation for the description of Coulomb interaction inside a bunch and its radiation in a relatively narrow waveguide.

_{ph}, and group, v

_{gr}, velocities of the wave are close to the speed of light, the longitudinal velocity of the particles changes, but these changes are small. The case of relatively low magnetic fields, where the lower intersection corresponds not to a backward, but to a forward wave with respect to the particle motion, is most interesting for radiation in the waveguide. Herein, the low frequency is not as different from that of a high-frequency wave as for the backward wave radiated by the plane. Correspondingly, the fraction of radiation into the low-frequency waveguide mode is much larger. The considered high-frequency mode for a bunch is similar in its properties to that used as the operating one for long beams in the so-called cyclotron autoresonance masers (CARMs) [20,21]. It is well known that CARMs are very critical to the spread in particle parameters and to the excitation of low-frequency waves with a small group velocity.

_{11}mode, the transverse fields are related by ${H}_{+,w}=i\frac{{k}_{z}}{k}{E}_{+,w}$, $\frac{{k}_{z}}{k}={\beta}_{gr}$. Therefore, the change in the particle axial momentum is

_{11}mode at the field H

_{0}= 22 kOe and emit in this case a broadband pulse with a center frequency of about 0.4 THz and a relatively high efficiency of about 10% (Figure 5).

## 4. Undulator Radiation of Electron Bunches in a Waveguide in the NM Regime

_{13}mode in Figure 9).

## 5. Conclusions

_{11}mode. It is important that with the initial longitudinal size of a bunch of the order of one-half of the corresponding wavelength even at relatively large transverse particle velocities, such a bunch weakly radiates at high harmonics s > 1 into the modes with azimuthal indices m = s. In a wider waveguide, the same bunch also radiates with high efficiency basically at the fundamental harmonic, but in a series of dominant modes with azimuthal indices m = 1. In this case, the highest radiation efficiency is achieved for the high-frequency mode with the lowest group velocity.

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 1.**Perfect 1D source of Coherent Spontaneous Radiation (CSR) in the form of a moving plane formed by uniformly distributed electrons, which synchronously move along identical helical trajectories and emit plane waves in the positive and negative directions of the z axis.

**Figure 2.**Cyclotron radiation of a moving plane: (

**a**) change in electron energy; (

**b**) current radiation frequency and efficiency for an initial particle energy of 6 MeV, normalized transverse velocity ${\beta}_{\perp 0}={\frac{1}{\gamma}}_{0}$, surface density $\sigma =1.2\times {10}^{-4}C/{m}^{2}$, and magnetic field ${H}_{0}=$27 kOe,${H}_{u}=0$.

**Figure 3.**Undulator radiation of the moving plane: current radiation frequency and efficiency for an initial particle energy of 6 MeV, transverse velocity ${\beta}_{\perp 0}={\frac{1}{\gamma}}_{0}$, surface density $\sigma =1.2\xb7{10}^{-4}C/{m}^{2}$, amplitude of undulator field ${H}_{0}=40.5\mathrm{kOe}$, and ${H}_{u}=$ 1 kOe.

**Figure 4.**(

**a**) The cross section of a cylindrical waveguide in which an electron bunch moves along helical trajectories; (

**b**) dispersion characteristic for the cases of the electron velocity is close to the group velocity of the radiated wave (“grazing”, G), and for the two waves at low (L) and at high (H) frequencies generation regime.

**Figure 5.**Cyclotron CSR of an electron bunch with a diameter of 1 mm, an initial duration of 0.25 ps, a charge of 0.1 nC, and a particle energy of 6 MeV in the group synchronism regime with the TE

_{11}mode: (

**a**) radiated energy, (

**b**) emission spectrum, and (

**c**) change in the rms durations for all the bunch and for its central part (without a long tail of particles).

**Figure 6.**Cyclotron CSR of a bunch with a diameter of 1 mm, an initial duration of 0.25 ps, a charge of 0.5 nC, and a particle energy of 6 MeV in the regime of simultaneous excitation of low- and high-frequency waves: (

**a**) radiated energy, (

**b**) emission spectrum, and (

**c**) change in the rms bunch durations.

**Figure 7.**Undulator CSR of a bunch with a diameter of 1 mm, an initial duration of 0.25 ps, a charge of 0.5 nC, and a particle energy of 6 MeV in the group synchronism regime with the TE11 mode of a cylindrical waveguide with a diameter of 4 mm: (

**a**) radiated energy, (

**b**) emission spectrum, and (

**c**) change in the rms bunch durations.

**Figure 8.**Undulator CSR of a bunch with a diameter of 1 mm, an initial duration of 0.25 ps, a charge of 0.5 nC, and a particle energy of 6 MeV in the regime of simultaneous excitation of low- and high-frequency waves: (

**a**) radiated energy, (

**b**) emission spectrum, and (

**c**) change in the rms bunch durations.

**Figure 9.**Multimode undulator CSR of a bunch with a diameter of 1 mm, an initial duration of 0.3 ps, a charge of 1 nC, and a particle energy of 6 MeV in the regime of simultaneous excitation of low- and high-frequency waves in a waveguide with a diameter of 10 mm: (

**a**) evolution of the particle density from the initial Gaussian distribution (black curves) to distributions after 10 and 40 undulator periods (blue lines) and (

**b**) emission spectrum for the undulator period d= 2.5 cm, undulator field amplitude ${\mathit{H}}_{\mathit{u}}=\mathbf{2}kOe$, and longitudinal field ${\mathit{H}}_{\mathbf{0}}=75kOe$.

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**MDPI and ACS Style**

Bratman, V.; Lurie, Y.; Oparina, Y.; Savilov, A.
Capabilities of Terahertz Cyclotron and Undulator Radiation from Short Ultrarelativistic Electron Bunches. *Instruments* **2019**, *3*, 55.
https://doi.org/10.3390/instruments3040055

**AMA Style**

Bratman V, Lurie Y, Oparina Y, Savilov A.
Capabilities of Terahertz Cyclotron and Undulator Radiation from Short Ultrarelativistic Electron Bunches. *Instruments*. 2019; 3(4):55.
https://doi.org/10.3390/instruments3040055

**Chicago/Turabian Style**

Bratman, Vladimir, Yuri Lurie, Yuliya Oparina, and Andrey Savilov.
2019. "Capabilities of Terahertz Cyclotron and Undulator Radiation from Short Ultrarelativistic Electron Bunches" *Instruments* 3, no. 4: 55.
https://doi.org/10.3390/instruments3040055