This report talks about Application of the Balanced Hybrid Mode in Overmoded Corrugated Waveguides to Short Wavelength Dynamic Undulator

/SLAC

/UCLA

Bowden, G.B.

Chang, C.

Neilson, J.

Pellegrini, C.

Shumail, M.

SLAC

Tantawi, S.G.

UNT Digital Library

APPLICATION OF THE BALANCED HYBRID MODE IN OVERMODED CORRUGATED WAVEGUIDES TO SHORT WAVELENGTH DYNAMIC UNDULATORS M. Shumail#, G. B. Bowden, C. Chang, J. Neilson, S. G. Tantawi, SLAC, Menlo Park, CA 94025 C. Pellegrini, UCLA, CA 90095, U.S.A.Abstract Inspired by recent developments in low-loss overmoded components and systems for ultra-high power RF systems, we explored several overmoded waveguide systems that could function as RF undulators. One promising structure is a corrugated waveguide system operating at the balanced hybrid HE11 mode. This is a new application for that mode. Initial calculations indicate that such a system can be operated at relatively low power levels (~50	MW) while obtaining large values for the undulator parameters ~1. RF surface fields are typically low enough to permit superconducting operation. This technology can realize an undulator with short wavelengths and gives dynamic control of the undulator parameters including polarization. We introduce the scaling laws governing such structures, and establish the superiority of the balanced hybrid HE11 mode by comparative analysis. The scaling laws are verified through exact simulations. High quality factor of HE11 mode is demonstrated through measurements of a prototype design. Finally, a novel design is presented with low surface fields. INTRODUCTION The dynamic RF undulators promise to impart a convenient control over the polarization and wavelength of the FEL radiation. Moreover, the large transverse radius of the overmoded waveguide weakens the effect of the wakefields [1-5]. In our analysis, we assume an undulator to be in the form of a circular cylindrical RF cavity, with either a smooth conducting wall or a finite constant impedance wall, of length  and radius . The device lies symmetrically about the z-axis along which we expect the electron beam to travel. Following notation is used for various RF parameters:  frequency ( ), angular frequency ( = 2 ), free space RF wavelength ( ), propagation constant along z-axis ( ), and free space wavenumber = . The normalized length is defined as = . However, to facilitate the comparison among various modes, we define the normalized radius as = ≥ 1, where  is, exactly (for TE1n) or approximately (for HE1n), the mode cutoff radius. In terms of the normalized radius, the propagation constant can be expressed as: = 	 	 √1 −                                                         (1) The modes of interest for any on-axis RF undulator, with cylindrical symmetry, are TM1n, TE1n, and HE1n modes.  For any given set of parameters, TE1n modes turn out to be superior to TM1n modes. In this paper, we present the scaling laws for the linearly polarized TE1n and HE1n modes. A comparison of these modes demonstrates the promising features of the HE11 mode. We also present the cold test results of a prototype corrugated cavity, and show that a high quality factor ( ) can be achieved for a HE11 mode.  In the end, we present our recent design of an RF undulator that exhibits low surface fields on the end walls. ZERO-IMPEDANCE-ADMITTANCE SURFACE FOR BALANCED HYBRID MODES For balanced hybrid modes, the impedance and admittance of the constant impedance wall should be vanishingly small [6-7]. In this case, the transverse field components of HE1n modes become proportional to , the Bessel function of order 0.  This gives a unique feature to the balanced hybrid HE1n modes that all its transverse fields are more concentrated on the axis and tend to vanish near the surface. Such a surface can be implemented by transverse corrugations of a metallic cylindrical wall, as shown in Fig. 1.   Figure 1: Longitudinal cross-section of a constant impedance cylindrical surface. In Fig. 1, the corrugation period  is less than 2⁄ , hence the slots can only support TM fields with field components,  , , and . Thus, at = , = 0, while there is a finite magnetic field . Thus the normalized surface impedance = − 	 →  vanishes. Moreover, it can be shown, that for ≫ 1, the normalized surface admittance = 			 	 → almost equals −cot Δ  [6] and hence vanishes for slot depth ≈ . In this way a so called zero-impedance-admittance surface is obtained. In practice, the value of < 2⁄  is selected to yield resonant condition for the  ___________________________________________ _____________ #shumail@slac.stanford.edu SLAC-PUB-15319Presented at the 2nd International Particle Accelerator Conference (IPAC 2011)San Sebastian, Spain, September 4 - 9, 2011Work supported by US Department of Energy contract DE-AC02-76SF00515.desired frequency. The value of  can then be tuned around  for optimal results, like lower power loss and surface fields. The slot thickness  needs to be just large enough to maintain the minimum aspect ratio for mechanical considerations. Generally, =  is a good choice. SCALING LAWS In the context of insertion devices, it seems appropriate to define following scaling factors for fields, energy, and power in terms of the undulator  parameter , normalized RF wavelength λ = λ 1	cm⁄ , and normalized conductivity σ (= 1 for copper). E = 4 	 	 	 	 = 642	MV	m 	 	 	 	                 (2) B = − E c⁄ = − 	2140	mT		 	 	 	                 (3) U = 	 = 	14.5	Joules		m 			 	            (4) P = 8 		 	 		 . 	 	 	 	             						= 2.089	MW			 	 ⁄ 	 	 ⁄ 	      (5) Here  and  are, respectively, electron mass and charge,  is the speed of light in vacuum, and  is the free space permeability.  Balanced Hybrid HE1n modes Observing the fact that ≈ (zeros of ) for a Hybrid balanced HE1n mode, we define normalized radial coordinate  and normalized radius  as = 	  and  = 		 . For ≥ 2, we can write following approximate expressions for a hybrid balanced HE1n mode: Propagation constant: ≈ 	 	 1 −                 (6) Undulator wavelength: ≈                   (7) Cutoff radius: ≈ 		 	                             (8) Field components (phasor form with implicit ): ≈			 	 	                                (9) ≈			 	 	 	                  (10) ≈ − 1 −                 (11) ≈			 1 −                 (12) ≈			 1 −                 (13) ≈			 1 −                (14) Stored energy per unit length: ≈ +   					≈ 	 − 0.25                                                (15) End wall losses: ≈ 	 −   																≈ 	 − 0.25 − 		                         (16) Cylindrical wall losses: ≈ 	 	 	 ≈ 2 	 	      (17) Peak surface fields: _ ≈ 	                                (18) _ ≈ 	                                            (19) In Eq. (15) and (17),  ζ ~1 and ζ ~1 are adjustment constants, and their values depend on the particular implementation of the zero-impedance-admittance cylindrical wall of the cavity. TE1n modes  For the sake of comparison, we present here the corresponding expressions for the stored energy per unit length and the losses for TE1n modes. Here, the normalized radius a is defined with respect to the cutoff radius of the corresponding TE1n mode. = 	 	 	 − 1		       ≈	 	 	 2 − 0.68 			                                         (20) 	= 	 − 1 	 	 − 1		                  ≈	 	 − 1 	 2 − 0.68                        (21) = 	 	 	                    + 1	 −   															≈ 2	 	 	 	 + 1	 −    (22)  Balanced Hybrid HE1n versus TE1n modes Comparing Eq. (15 – 17) with Eq. (20 – 22), one can guess that the HE1n modes do better than the corresponding TE1n modes. Fig. 2 presents a comparison of TE1n and HE1n modes, in terms of the stored energy ( ) and the total  losses ( = + ).    Figure 2: Stored energy ( ) and total losses ( ) versus normalized radius ( ). 1 2 3 4 5 6 70100200300400500aUJoulesHE11HE12HE13TE11TE12TE131 2 3 4 5 6 7050100150200aP lossMWHE11HE12HE13TE11 TE12TE13Here, the TE1n modes correspond to a smooth wall cavity, while the HE1n modes correspond to the cavity whose cylindrical surface is a constant impedance wall determined by the parameters: = 1 + 	 , = , and = . The modes are compared against the same values of their respective normalized radii. We have chosen: = = 1, 	 = 11.424	GHz, and = 1m. The dots correspond to the exact solution based on mode matching technique [8], while the solid lines are the plots of corresponding analytical expressions, Eq. (15 – 17) and Eq. (20 – 22) . We choose ζ = 0.38 and ζ = 0.92 to fit the HE11 data. It is obvious that the stored energy and the total power loss associated with the HE1n modes are generally lower, at least for this particular implementation, than that of the corresponding TE1n modes. In particular, HE11 mode outperforms all the rest, especially due to the lower associated power loss. PROTOTYPE TEST To validate our theoretical findings, we fabricated a prototype corrugated wall cylindrical cavity and  measured the quality factors ( ) of various modes. The design parameters of this cavity are as follows: ∆ =0.280	in, ∆ = 0.065	in, = 0.416	in,	 = 2.066	in, = 24 = 9.984	in, and = 0.431 (Aluminium). Table 1: Quality Factor for various Modes Mode Measured Calculated HE13 (11.279 GHz) 39,390 46,000 HE14 (11.308 GHz) 24,950 27,600 HE11 (11.428 GHz) 105,900 120,000 HE14 (11.507 GHz) 25,820 29,100 HE12 (11693 GHz) 67,100 85,600 Measurements confirm the high quality factor exhibited by the HE11 mode. However, all the measured values of  are about 10% less than the calculated values. The difference is probably due to the lower value of conductivity of the cavity alloy than that of the pure aluminum which is assumed for the calculated results. UNDULATOR DESIGN WITH LOW SURFACE FIELDS According to Eq. (3) and (19), the magnetic field on the end walls of the undulator can be as high as 815	mT for = 1, and = 2.6242	cm. To solve the problem of high peak fields on the end walls near the axis, we introduce a concave design for the end walls which is perpendicular to the local Poynting vectors. Fig. 3 shows such a design along with the plots of the magnitude of electric (E) and magnetic (H) fields along the axis of such an undulator. Such a design can help reduce the peak surface field by a factor of  half or so.  Figure 3: An optimized design of RF undulator and fields along its axis. CONCLUSION After a brief discussion about the design strategy of a zero-impedance-admittance wall to support the balanced hybrid HE1n modes, we presented scaling laws for these modes and compared them against the TE1n modes. It turns out that the HE1n modes have generally lower stored energy and exhibit less power loss than the corresponding TE1n modes. Especially, the HE11 mode seems to be the most promising one. We also presented results of a prototype design that exhibited high quality factor for the HE11 mode. Lastly, we presented a practical undulator design with low peak surface fields on the end walls.  REFERENCES [1] Tantawi, V. Dolgashev, C. Nantista, C. Pellegrini, J. Rosenzweig, G. Travish, “A coherent compton backscattering high hain FEL using an X-band microwave undulator”, FEL Conference 2005, Stanford, California, USA, 2005. [2] T. Shintake, K. Huke, J. Tanaka, I. Sato and I. Kumabe, “Development of microwave undulator”, Japanese J. of Appl. Phys., 22, p. 844-851, 1983. [3] T.M. Tran, B.G. Danly, J. S. Wurtele, “Free-electron lasers with electromagnetic standing wave wigglers”, IEEE J. of Quantum Electronics, Vol. QE-23, pp. 1578-1589, 1987. [4] M.Yeddulla, H. Geng, Z. Ma, Z. Huang, S. Tantawi, “Waveguide structures for RF undulators with applications to FEL and storage rings”, Proceedings of EPAC08, Genoa, Italy, 2008. [5] S. C. Pellegrini, “X-band microwave undulators for short wavelength free-electron lasers”, FEL Conference 2005, Stanford, California, USA, 2005. [6] P. J. B. Clarricoats, A. D. Olver, “Corrugated horns for microwave antennas”, Peter Peregrinus Ltd, London, UK, 1984. [7] S.F. Mahmoud, “Electromagnetic Waveguides theory and applications”, Peter Peregrinus Ltd, London, UK, 1991. [8] J. Neilson, P. Latham, M. Caplan, W. Lawson, “Determination of the resonant frequencies in a complex cavity using the scattering matrix formulation”, IEEE Trans. Microwave Theory and Tech., vol. 37, No. 8, 1989. 

Application of the Balanced Hybrid Mode in Overmoded Corrugated Waveguides to Short Wavelength Dynamic Undulators

Application of the Balanced Hybrid Mode in Overmoded Corrugated Waveguides to Short Wavelength Dynamic Undulators

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