Coupling impedance in modern accelerators

Using Parabolic Equation for Calculation of Beam Impedance

“…The step-in transition does not generate impedance in the limit of high frequencies which is also a well known result [5]. Note that solving PE for the step-in geometry we ignore the nonzero tangential electric field at the vertical part of the wall at z = 0.…”

“…This is a well known result for the longitudinal impedance for the step-out in the limit of high frequencies [5]. This impedance is real and corresponds to the energy loss of the beam when it builds up the electromagnetic field in the region of the expanded pipe.…”

“…[5] we will call the case b > a the step-in transition, and the opposite case the step-out, see Fig. 1.…”

“…An analytical expression was derived for the impedance of a periodic system of irises [9][10][11][12] which represents a long disk-loaded accelerating cavity. Attempts were made to generalize the diffraction theory for other geometries [5] based on an integral equation approach, however, the method of reference [5] works only for some special geometries and, as one can show, does not give the correct numerical factors in the expressions for the impedance.…”

“…Several analytical results are available in the literature for the impedance in the high-frequency limit. They include the impedance of a step transition [5] and the diffraction model for the impedance of a cylindrical cavity [6][7][8][9] when the length of the cavity satisfies the relation l cav ≪ b 2 /λ with b the radius of the entrance pipe to the cavity. An analytical expression was derived for the impedance of a periodic system of irises [9][10][11][12] which represents a long disk-loaded accelerating cavity.…”