HOM power propagation and attenuation in PEP-II B-factory

Most of the higher-order-mode (HOM) power that propagates in the PEP-II rings is generated in the RF cavities but its content in terms of TE and TM components has not been accurately determined. For purpose of shielding beamline components such as bellows and BPMs from TE power penetration and determining the heating on the cavity HOM absorber, this HOM power content and its distribution profile around the rings are needed. This paper calculates the TE and TM contributions of the RF cavities to the circulating HOM power and their transmission properties at another cavity downstream. By considering both the generation in, and scattering by the cavities, as well as the attenuation along the vacuum chamber, a complete distribution profile for the TE and TM HOM power can be obtained. 1 HOM POWER GENERATION IN PEP-II The PEP-II B-factory is a high current machine that generates enormous amount of RF energy in the vacuum chamber. The total HOM power for all beamline elements is estimated to be P0 = 200 Kilowatts[1]. Assuming that the HOM power is uniformly generated around a storage ring and the attenuation length l is constant for all the propagating modes of the vacuum chamber, the HOM power P at any location of the ring is given by the equilibrium condition dP=dz + P0=L = P=l+ P0=L = 0; (1) where L, the PEP-II ring circumference, is 2200 m. The straight sections of the PEP-II rings consist of stainless steel pipes with a diameter of 9.525 cm. The attenuation length of the TM01 mode in these pipes is about 60 m at 5 GHz, the roll-off frequency of the PEP-II nominal 1 cm bunch. Thus the resulting HOM power at any ring location is roughly given by P = l L P0 = 5:5 KW: (2) 2 MULTIPLE REFLECTIONS The above calculation assumes no reflections by beamline elements. On the contrary, multiple reflections may happen even in a two-element network. To illustrate the significance of multiple reflections, we study the cascade of two simple 2-port elements as shown in Fig. 1. Here, S represents the scattering matrix of the two identical elements, Work supported by U.S. Department of Energy, contract DE-AC0376SF00515 e a iα S S 1 Tt Rt b Figure 1: Cascade two 2-port elements with one unit of power incident from left. and e indicates the propagation through the beampipe between the two elements. The amplitudes of the right-going and left-going waves between the two elements are represented by a and b, respectively. We parameterize the Smatrix in the form of S = r i p 1 r i p 1 r r ; (3) where r represents the reflection coefficient. The particular form is chosen to satisfy symmetry and unitarity. With one unit of power incident from the left, the total transmission Tt is obtained by summing over all the reflections as follows: Tt = Te i T (1 + re re (1 + re re (1 + :::))); (4) where T = S12 = i p 1 r. The first term is the direct transmission, and the second term has two extra reflections on the network interfaces and two extra phase propagations. Summing the geometric series, we obtain Tt = (1 r) e 1 re : (5) Following the same procedure, we also have