Nuclear Instruments and Methods in Physics Research A 359 (1995) 47-49 ELSEVIER zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA NUCLEAR INSTRUMENTS B METNoDs IN PNYSICS REWASH An undulator for a FEL based on a solenoid with superconducting diaphragms V.A. Bordovitsyn a, V.Ya. Epp b, * , A.V. Kozhevnikov ‘, V-F. Zalmezh ’ ’ zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA Tomsk State Universi& 634050 Tom&, Russian Federation b Tomsk Pedagogical Institute, 634041 Tom& Russian Federation ’ Tomsk Polyiechnic University, 634004 Tomsk, Russian Federation zyxwvutsrqponmlkjihgfedcbaZYXWV Abstract Two types of magnetic undulator designs with superconducting diaphragms are presented. The magnetic field is simulated numerically and discussed qualitatively for one of them. For the another one the charged particle motion is analyzed. The discovery of high-temperature superconductors has made superconducting materials more available and lower in cost. The diamagnetic properties of superconductors allow us to use them to form the magnetic field of undulators. The idea of using a periodic assembly of rings for the purpose of shaping the undulator magnetic field has been expressed in Refs. [l-3]. Iron [1,2] and copper [3] rings have been applied to modulate the field of the solenoid. Two types of magnetic undulator design with supercon- ducting diaphragms are presented in this paper. i) A schematic sketch of the fit one is presented in Fig. la. The ring-shaped superconducting diaphragms of thickness b, inner radius rd and outer one r, are periodi- cally spaced in the solenoid with pitch 1 beginning at distance d away from the edge of the solenoid. The lines of the magnetic force skirt the inner edge of the di- aphragms because of the diamagnetic nature of supercon- ductors. Thus the longitudinal component B, of the mag- netic field takes its maximum value in the plane of the diaphragm. In the case of an axial-symmetrical field with the standard condition div zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCBA B = 0 fulfilled, the radial com- ponent B, can be expressed as a gradient of B, as follows: r aB, B,= -yy. The results of a numerical simulation of the magnetic field components B,(r) and B,(z) for the case of thin diaphragms (b/r, +z 1) with radius r,, = 0.4r, and period 1= rs are presented in Figs. lb and lc. Four values of radius r have been considered: r = 0.25r,, OSr,, 0.75r,, * Corresponding author. and l.Or,. One can see that the magnetic field components B, and B, increase nonlinearly with r taking its maximum at r = rd. They depend harmonically on z in the vicinity of the axis. But their behavior differs considerably at the edge of the gap near r = rd. We find out that with the increase of the diaphragm thickness b the modulation depth of B, and B, reduces, and their variation along z becomes similar to a sinusoidal one. One can place ferromagnetic rings between the di- aphragms for the purpose of realizing a short-period undu- later and keeping the relatively high amplitude of B,. Fig. 2 illustrates the variation of B, versus r with ferromag- netic material between the diaphragms (curve 1) and with- out it (curve 2). The ratio BJB, may be as much as 2 or even 3 if the ferromagnetic rings are used. This value far exceeds the modulation depth in the case of iron or copper rings used in Refs. [l-3]. The simulation results presented above show that the most suitable electron bunch is a tube-like one of low thickness and with a radius slightly below the diaphragm radius rd. It makes this kind of undulator favorable for free electron lasers based on high- current accelerators. ii) A short-period undulator dedicated to a tube-like beam has been suggested in Ref. [4] and is shown in Fig. 3. The magnetic field is set up by an electromagnetic solenoid (1) and a row of alternating ferromagnetic (2) and superconducting (3) rings. It is proposed that a supercon- ducting splitter should be installed in line with the undula- tor axis in order to concentrate the magnetic field in the beam region and to keep the beam close to the inner surface of the rings. The vector potential of the magnetic field in the interac- tion region is given by the Laplace’s equation AA = 0. The magnetic field of an infinitely long device can be 016%9002/95/$09.50 0 1995 Elsevier Science B.V. All rights reserved SSDI 0168-9002(94)01365-9 II. RADIATION SOURCES