Helical Undulator Based on Partial Redistribution of Uniform Magnetic Field

A new type of helical undulator based on redistribution of magnetic field of a solenoid by ferromagnetic helix has been proposed and studied both in theory and experiment. Such undulators are very simple and efficient for promising sources of coherent spontaneous THz undulator radiation from dense electron bunches formed in laser-driven photo-injectors.


I. INTRODUCTION
Advanced laser-driven photo-injectors make possible formation of very dense picosecond and sub-picosecond electron bunches with charge of the order of 1 nC and moderate energy of 3-6 MeV [1][2][3][4][5]. Such bunches can be attractive for production of power THz electromagnetic pulses [6][7][8][9][10][11][12][13], if very strong mutual Coulomb longitudinal repulsion of bunch particles is prevented or mitigated. One of the recently proposed efficient methods for stabilization of the electron bunches length and for corresponding increase of their spontaneous Doppler-upshifted terahertz undulator radiation energy [12,13] is based on development of the so-called Negative-Mass Instability (NMI) [14][15][16][17] in the bunches moving in undulators. Implementation of the method requires the undulator field to be combined with a strong uniform axial guiding magnetic field [12,13]. Here, the "strong" means that the value of the uniform guiding magnetic field should be higher than the resonant value corresponding to the coincidence of undulator and cyclotron frequencies of the electrons. In this paper, we demonstrate that in such a system, the undulator field can be easily created by redistributing the strong uniform field by a periodic ferromagnetic helix. A similar method was proposed and demonstrated many years ago in planar undulator systems with periodic planar insertions [18][19][20]. Also, helically distributed planar elements were used for creating a relatively low helical transverse magnetic field [21]. We suggest using a more convenient helical undulator based on a fully-symmetrical helical insertion [22]. It is fairly obvious that magnetization of a ferromagnetic helix in the guiding field provides a helical component of the magnetic field, and we will show that a sufficient for realization of the NMI-based terahertz source amplitude of this component can be readily obtained.

II. ANALYTICAL ESTIMATING
According to [12,13], development of NMI inside a dense electron bunch can lead to effective mutual at- * iluy@appl.sci-nnov.ru traction of particles. As a result, a dense core with a fairly stable longitudinal size smaller than the radiation wavelength can appear. Such longitudinal bunching occurs in a combined helical, B u , and over-resonance uniform guiding, B 0 z 0 , magnetic fields when the electron cyclotron frequency, ω c = eB 0 /mγ, is larger than its undulator bounce-frequency, ω u = 2πv /d. Here, e, m, γ and v are the electron charge, mass, Lorentz-factor, and longitudinal velocity, and d is the undulator period. In this over-resonance region of parameters, the increase of the energy, mc 2 γ, of a front electron under the action of Coulomb field of the rest of the bunch causes approaching of the electron to the resonance, ω c ≈ ω u . If the electrons move along their quasi-stationary helical trajectories, it is accompanied with increase of the transverse electron velocity, v ⊥ = cK/γ∆, where K = eB u d/2πmc and ∆ = 1 − ω c /ω u are the undulator parameter and the resonance mismatch. In certain conditions, the transverse velocity grows so quickly that the derivative ∂v /∂γ is negative, and the longitudinal components of the front bunch particles velocities decrease. Simultaneously, particles from the rear part of the bunch lose their energy and move away from the resonance, which is accompanied with the decrease of transverse and increase of longitudinal components of velocities. Thus, similarly to the classic cyclotron Negative-Mass Instability in cyclic accelerators [23,24] and gyrodevices [25][26][27], the change of the electrons energy in the repulsing Coulomb field leads to an effective longitudinal attraction and bunching of non-isochronously oscillating particles.
For typical parameters of a THz source, considered in [12, 13] (particle energy of 5-6 MeV, bunch charge of up to 1 nC, and radiation frequency of 1-2 THz), the value of undulator field, B u , should be of the order of 0.1-0.2 T, that is tens times smaller than the guiding field, B 0 . Correspondingly, a relatively thin steel helix is supposed to be sufficient for providing the required value of undulator field.
Let us find the field of a helical insertion placed into an infinite solenoid with a uniform field. Consider magnetization J of an infinite ferromagnetic helix with period (step) d, inner and outer radii R 1 and R 2 , respectively, thickness b = R 2 − R 1 , and axial size a (Fig. 1). In the case of a very strong guiding field, the magnetization of the helix is saturated, J = J ∞ , so that its direction prac- tically coincides with direction of the field, z 0 , and its value does not depend on B 0 .
Let us consider an elementary thin helix, for which R 1 ≈ R 2 ≈ R, b ≪ R, and introduce its surface magnetization. Taking into account that the helix bulk is periodic in axial direction and invariant to the transformations keeping ϕ + 2πz/d = const, we use the Fourier expansion and represent the magnetization in the form where h = 2π/d and f n = (1/πn) sin(nha/2). The nth harmonic of the magnetization, J n , presents a continuous polarized cylindrical surface with radius R; div J n can be interpreted as equivalent continuously distributed surface magnetic charge. The corresponding harmonic of the static magnetic potential satisfies the Poisson equation ∆ψ n = div J n .
Boundary conditions at the cylindrical surface r = R correspond to continuity of the potential and jump of the radial component of magnetic field strength H n = −∇ψ n : The solution of Eqs. (2), (3) is where I p , K p are modified Bessel and McDonald functions of the order of p = |n| and g n = hbRJ ∞ f n . It is easy to show that only harmonics n = ±1 in Eq. (4) contribute to the transverse undulator field at the axis of the system. The corresponding field is where Maximum of the transverse field is achieved when the longitudinal size of the helix is equal to half of its period, a = d/2. Integration of expression (4) over the radius gives solution for the helix of finite thickness.

III. NUMERICAL SIMULATIONS
The applicability of the solution (5)-(6) was verified by direct solving of magnetostatic equations using CST Studio. This code allows one to take into account a finite thickness, arbitrary transverse and longitudinal profiles of the helix as well as nonlinear dependence of magnetic permeability of steel on applied field. Numerical simulations demonstrate a high accuracy of the solution (5) admits quite well with 0.21 T given by the CST Studio simulations for these parameters and axial magnetic field of 8 T ≫ µ 0 J ∞ ≈ 2.15 T (Fig. 2).
Also, formula (6) can be well applied for thin helices with round cross-sections if one takes parameters a and b equal to each other, so that ab corresponded to the square of the wire cross-section, and R equal to the mean radius of the helix bulk (Fig. 3).

IV. EXPERIMENTAL MEASUREMENTS
The described method of creation of helical undulator field has been also studied experimentally for a helix of square cross-section (Fig. 4). In the experiment, using a probe in the form of a small coil, the transverse field was measured on the axis of the helix placed into pulsed fields with maximum values of 4-5 T. The probe coil contained ∼200 turns and had a length of 5 mm and an average diameter of 5 mm. To compensate for the inevitable deviation of the coil axis from the radial direction, leading to a contribution from the strong longitudinal field, the probe was rotated around the main solenoid (z) axis, and In the first experiment, we used a steel helix with the period of 2.5 cm, inner radius of 0.6 cm and square crosssection having thickness of 0.4 cm (Fig. 4a). The steel type was equivalent to ASTM-SAE 1045 having the carbon content less than 0.5%, which means that the saturation magnetic field was about 2 T [28]. The helix was manufactured from steel cylinder first by milling the cylinder surface and then by cutting away the inner part by electrical discharge machining. Being mounted in the pulsed solenoid with an axial magnetic field of about 4.5 T, it created the transverse helical field of the order of 0.1 T (Fig. 5). The main source of measurements error was connected with displacements of the probe coil from the solenoid axis, which were caused partially by misalignment of the probe support during the rotation of the probe, and partially by mechanical shakes caused by pulsed magnetic field of large amplitude. The difference in field values measured at different axial probe positions can be explained by the axial inhomogeneity of the solenoid field, which had the axial scale of about 10 cm. It should be noted, that in real application, such a helix is supposed to be placed inside a DC or long-pulse solenoid, whereas in the described experiment, the solenoid pulse had an almost sinusoidal form, B 0 ∼ sin(πt/T ), with a length of "half-period" T ≈ 3 ms. In principle, in the pulsed case, a second source of transverse magnetic field appears, which is the induction current along the helix. Nevertheless, simple estimations show that for these particular pulse duration, helix geometry and material conductance, the correspondent magnetic field is almost two orders of magnitude lower than the one induced by magnetization.
It is very important, that the simple formulas (5)- (6) give for the experimental parameters the very same value of transverse magnetic field, B u ≈ 0.09 T, and so do the CST numerical simulations. This proves the correctness of theoretical calculations and assures that the undulator field of 0.2 T, which was predicted by calculations of Section III and corresponds to desirable undulator parameter, K ≈ 0.5, indeed can be achieved experimentally.

V. CONCLUSION
The proposed method of creating a helical undulator field by redistributing the uniform field at a steel helix seems very simple and efficient. It can be naturally applied to THz sources of coherent spontaneous undulator radiation using the NMI stabilization of longitudinal sizes of short picosecond and sub-picosecond electron bunches [12,13]. According to simulations, development of NMI in a combined strong uniform and helical undulator fields can provide maintenance of nearly constant sizes of short and long-living bunch cores at a fairly large undulator length. It makes possible the implementation of powerful and narrowband radiation sources with initial electron energy of about 5-6 MeV and charge of the order of 1 nC with the frequency of 1-2 THz and efficiency of about of 20%.