Spatial harmonics of linear multipoles with round electrodes

The problem of creating hexapole, octopole, decapole and dodecapole electric fields with round-rod electrodes is considered. We propose a new numerical method to calculate the spatial harmonics and find the optimal electrode configurations. This configuration is characterized by the parameter gamma = r/r(0), where r is the rod radius and r(0) is the radius of an inscribed circle between the electrode tips. We consider four different criteria for optimizing the field: (1) the value that makes the amplitude of the main multipole 1.0, (2) the value that makes the amplitude of the next higher harmonic after the main harmonic zero, (3) the ratio that gives the next two higher harmonics equal amplitudes but opposite signs and (4) the ratio that minimizes the deviation of the potential and electric field from the potential and electric field of an ideal multipole, averaged over the region within the multipole. Each gives slightly different values for the optimal value of gamma. To minimize the field deviation from that of an ideal multipole the optimal values are gamma = 0.563, 0.372, 0.278, 0.221 for hexapole, octopole, decapole, and dodecapole fields, respectively. (C) 2009 Elsevier B.V. All rights reserved.