Abstract
The magnetic field of a staggered-array undulator using a bulk high-temperature superconductor is calculated by analytical and numerical methods. Analytical formulas for the undulator field and the solenoid field required to generate the undulator field are derived from a simple two-dimensional model. The analytical calculation shows the degree of dependence of these fields on the undulator parameters, the generation of a high undulator field proportional to the critical current density of the bulk superconductor, and the good tunability of the undulator field over a wide range of values. The numerical calculation is performed in a three-dimensional geometry by two methods: the center field and energy minimization methods. The latter treats the current distribution inside the bulk, whereas the former neglects it as a natural extension of the analytical model. The calculation also reveals the dependence of the fields on the undulator parameters arising from the current distribution. From the comparison with experimental results, we find that the latter method reproduces the experimental results well, which indicates the importance of the current distribution inside the bulk. Therefore, we derive a semiempirical formula for the required solenoid field by modifying the analytical formula using the numerical results so as to include the effect of the current distribution. The semiempirical formula reproduces the numerical result with an error of 3%. Finally, we estimate the magnetic performance of the undulator as an example of using the formulas and values presented in this paper. The estimation shows that an undulator field twice as large as that of the present in-vacuum undulator but with an equal period and gap can be obtained at a temperature of approximately 20--40 K, and that deflection parameters ($K$ values) of 1 and 2 can be achieved with periods of 5 and 10 mm at approximately 4--20 K.
Highlights
Synchrotron radiation from a relativistic electron beam passing through an undulator, i.e., undulator radiation, and free electron lasers (FELs) are crucial to a wide range of scientific applications
The analytical calculation showed that a high undulator field could be achieved by the bulk HTS staggered array undulator (BHSAU) at a high critical current density and that good undulator field tunability was maintained even in the saturation region
The numerical calculation by the energy minimization (EM) method, which treats the current distribution inside the bulk high-temperature superconductor (HTS), produced a considerably different current distribution from that obtained by the center field (CF) method, and closely reproduced the experimental results
Summary
Synchrotron radiation from a relativistic electron beam passing through an undulator, i.e., undulator radiation, and free electron lasers (FELs) are crucial to a wide range of scientific applications. (3 GeV, 21 mm), and SOLEIL in France (2.75 GeV, 20 mm), have been constructed [2,3,4] In these facilities, the fundamental wavelengths of undulator radiation reach the x-ray region; harmonics from the 5th through 11th reach the hard x-ray region. The fundamental wavelengths of undulator radiation reach the x-ray region; harmonics from the 5th through 11th reach the hard x-ray region This scheme prevails for FELs. The first x-ray laser of 0.12 nm (10 keV) was achieved at LCLS in the U.S (15 GeV, 30 mm) [5], and such lasers have already been used for x-ray science [6]. As can be concluded from the above, short-period undulators require lower electron beam energies, meaning lower costs, a shorter construction time, and
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