New Method to Monitor the Transverse Distribution of Current in Particle Beams

A group of sinusoidally-wound coaxial toroidal coils can be used to determine the transverse distribution of a time-dependent current through their common aperture. The current is expressed in a basis of chapeau (pulse) functions over an array of pixels, and matrix methods are used to calculate the current in each pixel from the voltages induced on the coils. Optimum configurations of pixels are used, for which the condition number of the matrix is equal to the number of pixels. For example, with 50 pixels the fractional errors in the currents are 50 times the fractional errors in the measured voltages in addition to imperfections in the fabrication and placement of the coils. Numerical tests were made by specifying the currents, calculating the induced voltages, adding Gaussian noise to model the measurement errors, and then using the algorithms to calculate the currents. These simulations confirm that the condition number is equal to the number of pixels. Means for constructing prototypes are also described.