Abstract
We employ Abel's transform to derive an analytic expression for filtering an equilibrium distribution function from its longitudinal beam profile. The result can be applied to a large class of Hamiltonians that are quadratic in the momentum coordinate.
Highlights
Longitudinal phase space tomography is developing rapidly into a practical diagnostic technique
Schlarb [4] described a procedure used at DESY, based on a thesis of Geitz [5], for reconstruction of transverse distributions
The purpose of this paper is to present an analytic expression for obtaining an equilibrium distribution function from a beam profile, an integral transform whose numerical implementation is equivalent to inverting that triangular system of equations
Summary
Longitudinal phase space tomography is developing rapidly into a practical diagnostic technique. In either case one obtains a lower triangular linear system of equations which must be inverted, not an unusual occurrence in tomographic problems. The purpose of this paper is to present an analytic expression for obtaining an equilibrium distribution function from a beam profile, an integral transform whose numerical implementation is equivalent to inverting that triangular system of equations. This is much like an integral equation motivated by one of those ‘‘bead on a frictionless wire’’ problems: Suppose that a mass drops in a constant gravitational field along a contrained path, expressed as the set of points f yz; z 2 E2 jz 2 0; 1g: Let the time that it takes for the bead to reach z 0 when it is dropped from an altitude z h be represented by the function h. The problem is expressed as an integral equation by using conservation of energy: p mv mgz mgh ) v 2gh z means that
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