Phase-space measurement of a beam with a Maxwellian transverse velocity distribution by a finite-width slit

The beam is best represented by its distribution in the transverse phase space. In order to determine the transverse velocity distribution, either a pinhole or a thin slit may be used. The slit, however, is more often preferred in many experiments I4 for two,-dimensional phase-space measurement. A beam produced by a hot thermionic cathode or a beam which has existed in a space for a long period of time tends to have a Gaussian distribution. It is shown4 that for a beam with Gaussian distribution, the rms emittance can be analyzed after data obtained with a slit system. The slit samples the beam at a given slit location producing a sheet beamlet. The beamlet is then allowed to disperse according to its transverse velocity distribution. The beam velocity distribution can be found from the beamlet density profile measured at the detector plane farther downstream of the slit. Ideally, an infinitely thin slit should be used, which would produce a density profile that is equivalent to that of the original velocity distribution of the beam. In actual experiments, however, use of a finite-width slit is inevitable for the following reasons: a narrower slit reduces the beam intensity, resulting in a poorer signal to noise ratio of data; the minimum slit width often is limited by a fabrication technique employed; the narrower slit with a finite thickness of slit plate restricts the allowed acceptance angle of the beam. In this article, we describe a convenient method of obtaining the original Maxwellian distribution from the density profile of a finite-width slit produced beamlet. In general, the profile is deformed from the original beam distribution due to the finite slit-width. Force-free drift motion of particles in the space between the slit and the detector planes is assumed. The density profile of the beamlet on the detector plane can be expressed by two error functions.3 The rms velocity width is found as a function of the slit width and the measured width of the sheet beamlet’s density profile that is produced by the slit. The peak value of equivalent sheet beam distribution is found in terms of measured peak value and the slit width. Although such relations are solved numerically, the results are leastsquares fit to polynomials with a high degree of accuracy so that actual experimental data can be conveniently analyzed.