Permanent Helical Undulators with Strong Fields

Undulators containing magnetized rare-earth helices can provide a significantly higher oscillatory electron velocity than the widely used planar Halbach undulators. Using Wire Electrical Discharge Machining (WEDM), it is possible to manufacture NdFeB helices with a period of 1 mm or less with high accuracy. In this work, we describe the results of manufacturing and studying prototypes of undulators in the form of one or two axially and radially magnetized helices. More efficient hybrid systems of two axially oppositely magnetized and two steel non-premagnetized helices with a field on the axis of the order of 1 T are also shown. Micro-undulators of this type can significantly increase the efficiency of XFELs and Inverse FELs.


Permanent Helical Undulators from Helices
One of the key elements in all projects of compact XFELs and IFELs are high-field micro-undulators.For example, in the XFEL-UC concept [1], this is a micro-undulator with a period of down to 3 mm and a field of 1 T. To do this, it is supposed to use a planar structure of Halbach lattices [2], assembled from solid combs of equally magnetized elements [3].For the same purpose, we consider Halbach-type helical undulators of helices, each of which, like the combs in [1], is made of a single piece of ferromagnetic material [4].Compared to a planar system, with the same gap for electron transport, a helical undulator provides a significantly larger rms oscillatory velocity and, consequently, a higher radiation efficiency or particle acceleration at a shorter length.
An obvious generalization of the planar Halbach undulators [2] to the case of helical geometry is a set of four alternating axially and radially magnetized helices with a width of a quarter of the undulator period d each (Fig. 1) [4].The magnitude B of a helically polarized field on the axis of such undulator formed by infinite helices of a rectangular cross section is [4]  function and its derivative, ℎ = 2/,  1,2 = ℎ 1,2 ,  1,2 are the inner and outer helix radii.
As in planar undulators, the helical system provides a stronger field on the axis than the simplest system of two axially magnetized helices and a weak external field.If the thickness of the magnets,( 2 −  1 ), is large enough and the gap between them, 2 1 , is small, the field on the axis is close to the remnant field.For example,  = 0.94  in the infinite planar system and  = 1.10  in the infinite helical system with the same dimensionless parameters  1 = 0.2and 2 = 1.5.For NdFeB magnets with a typical value   = 1.2, both values exceed 1T.At the same time, with equal field magnitudes, the proposed helical undulator system provides an oscillating electron velocity with a √2 times larger rms.A helical undulator in the form of a helix assembled from many small magnets was mentioned in [2].We assume that the development of this idea in the proposed undulator, consisting of helices made of solid pieces, can provide the required periodic field distribution and easier manufacturing with a better accuracy, especially in the case of a short period.It is not easy to obtain the radial magnetization of a helix.However, for this purpose, as with planar undulators, one can use simpler hybrid systems consisting of alternating oppositely axially magnetized helices and steel helices between them, which redirect a magnetic flux and provide radial direction of magnetization at the inner insert surface (Fig. 2).Optimization of such system allows a 15% increase in the field on the axis compared to a system with the same width of helices [5].High-quality NdFeB helices without inner holes and with periods down to 1 mm intended for other purposes have already been successfully manufactured using WEDM [6] Making the helix from one piece can also reduce the effect of possible deviations of the undulator field caused by the inhomogeneities of a rare-earth material and the directions of magnetization of its numerous individual elements.By analogy with a planar system of magnetized combs [3], to weaken the strong repulsion of oppositely magnetized and brittle NdFeB helices, when assembling the proposed design, a system of four combs but with spiral elements can be used (Fig. 3).

Helical Micro-Undulators for Compact FELs
Let us compare the proposed hybrid Halbach-type helical system as a micro-undulator for a compact Xray FEL with a planar Halbach micro-undulator from comb elements considered in [1,3].Table 1 lists several key parameters of electron bunches and a planar Halbach micro-undulator considered in the design of an ultra-compact XFEL with an electron energy of 1.6 GeV, an undulator period of 3 mm, and a radiation wavelength of 1.6 Å [1].
According to calculations for Br =1.4 T, the same field amplitude 1 T as in a planar micro-undulator with a period of 3 mm and a gap of 0.88 mm, but in both transverse polarizations of the circularly polarized field, can be achieved in the helical micro-undulator with an inner diameter of 1.06 mm.According to GPT calculations, a single particle with initial coordinates  = r u ,  = 0, injected on a quasistationary helical trajectory, oscillates with the amplitude   ≈  ⊥ /  = 0.04 at the fast undulator frequency   = ℎ  .It simultaneously participates in slow betatron oscillations, deviating to distances  ≈ −0.05μm,  ≈ 0.05μm  on 600 undulator periods.However, a particle with the same initial coordinates deviates to a much greater distance in the fields of the micro-undulator and space charge of the bunch with the radius, length, and charge indicated in Table 1.Correspondingly, the transverse size of the bunch changes drastically over the length of particles microbunching in an XFEL.
To prevent transverse expansion, additional focusing of the electron bunch is certainly necessary, as is provided, for example, by a periodic system of quadrupoles with a very strong field gradient in the case of a planar Halbach micro-undulator in [1].Using the GPT code, we selected the same parameters of quadrupoles for focusing a bunch moving in helical and planar micro-undulators with 1-T fields on the axis, namely, a gradient of 350 T/m and a length of 4 cm of quadrupoles located 0.3 m from each other.Up to the difference in the amplitude of undulator oscillations such a system provides the same average beam radius and the same level of spread in longitudinal particle velocities in both types of micro-undulators (Fig. 4).
Let us compare the main parameters of the FEL instability for the cases of planar and helical microundulators.According to the 1-D theory (see, e.g., [8, 9, 1]), the Pierce parameter and exponential gain length are Here, I is the bunch current, IA is the Alfven current,  and  are the transverse bunch rms size and the radiation wavelength, respectively, and K is the undulator parameter.
Figure 4. Almost identical envelopes for a bunch with the parameters from Table 1 in helical and planar microundulators with a period of 3 mm and a field of 1T on the axis and periodic quadrupole focusing.
At the same field amplitude B, the parameter K2 for a helical undulator is twice as large as for a planar one.Thus, for the same focusing of particles, replacing a planar undulator with a helical one increases the Pierce parameter and the radiation efficiency by a factor of √2 3 = 1.26 and reduces the length of the radiation section by the same factor.

Fabrication and Experimental Study of NdFeB Helices
A non-magnetized NdFeB type N50 cylinder with an increased remnant magnetization Br =1.4 T, an outer radius of 15 mm, an inner radius of 4 mm, and a height of 40 mm was used to make two helices with a period of 20 mm.The WEDM was used to simultaneously cut the cylinder into two desired helices with high precision and material damage only in a very thin layer (Fig. 5).The longitudinal magnetization of the helices was carried out in a pulsed solenoid with a field of more than 2 T and a pulse duration of 2 ms.The measured field of the magnetized helix agreed very well with the CST simulation (Fig. 6).As the next steps, we are going to fabricate and assemble relatively long hybrid undulators of axially magnetized NdFeB helices and non-premagnetized steel helices with periods of 20 mm and 3 mm, designed accordingly for use in terahertz FELs and to demonstrate the possibility of using such systems to implement ultracompact XFELs.

Conclusions
We propose an alternative version of a permanent micro-undulator, which can be used to create an ultracompact X-ray FEL, the design of which is presented in [1].To manufacture a planar micro-undulator assembled from magnetized NdFeB combs, in [1,3] it is pro-posed, in particular, to use the Wire Electric Discharge Machining.The same technology makes it possible to  manufacture the proposed helical micro-undulator consisting of magnetized NdFeB helices with a micron accuracy.This technology has already been successfully employed to manufacture helices with periods down to 1 mm [6,7] but without internal axial holes for transporting electron bunches.In this work, we have demonstrated the possibility of using this technology to fabricate a short undulator prototype with a relatively large period of 20 mm and a strong field.The micro-undulators of helices can significantly increase the energy capabilities of compact sources of high-power coherent X-ray radiation proposed in [1] and many other papers.

2 Here,
Figure 1: A Halbach-like Helical Undulator Comprising Four Helices with Axial and Radial Magnetizations.

Figure 2 .
Figure 2. (a) A hybrid undulator with alternating axially pre-magnetized NdFeB helices and non-magnetized steel helices, (b) the field found from CST simulations.

Figure 3 .
Figure 3.A hybrid undulator consisting of combs with rare-earth helical elements and steel helices: (a) assembled system, (b) the combs are separated, and (c) a front view of the separated system.

Figure 5 .
Figure 5. Two identical helices obtained simultaneously from WEDM thin spiral cut of a NdFeB cylinder.On the left is support with a spiral track from WEDM.

Figure 6 .
Figure 6.Comparison of measurements and CST simulations for the field of a longitudinally magnetized helix with a period of 2 cm and an inner radius of 4 mm.