Measurement of Pulsewidth via Correlations in Intensity Fluctuations

We conisder a novel method to measure the width, ∆t, of a pulse of relativistic electrons (a beam pulse), via the correlations in the intensity fluctuations of radiation emitted when the pulse passes through a magnetic field. Since intensity fluctuations appear to be a kind of noise, this technique is somewhat counterinituitive. The measurement is based on synchrotron radiation, whose spectrum extends up to a maximum angular frequency ωmax À 1/∆t. This behavior indicates that the radiation is not (first-order) coherent. Then, we can examine the frequency spectrum around a central frequency, ω0 À 1/∆t. At this high frequency, we must take into account the phase difference between light emitted by different electrons in the beam pulse. Indeed, if the electrons were arranged on a lattice, the phase differences would result in essentially complete destructive interference, and there would be no signal. But because of fluctuations in the positions of the electrons, a useful signal results. For n electrons, the average intensity at frequency ω0 À 1/∆t is n times that from a single electron; the rms amplitude is √ n times that for a single electron. The radiation at such frequencies is incoherent, in contrast to coherent radiation for which the intensity is n times that for a single electron. The amplitude of the radiation at a particular frequency could, of course, be positive, or negative, or very close to zero. Thus, if we examine the radiation spectrum over a range of frequencies near ω0, the amplitudes will vary over the range ±√n; the intensity will vary between 0 and n (times that due to a single electron). There are 100% fluctuations in the intensity as a function of frequency, and the itensity spectrum appears noisy. In optics, a source whose intensity is n times that of a unit source, and whose intensity fluctuations have rms magnitude n, is called a “thermal” or “chaotic” source, as first described by Rayleigh [1]. In case of a signal based on n independent samples, one class of fluctuations will have size √ n. The 100% intensity fluctuations that arise here should perhaps be called fluctuations of the fluctuations, as discussed in greater detail in sec. VI. The light from a thermal source, although described as incoherent, still manifests intensity correlations that contain information as to the temporal pulsewidth of the source, which can be extracted with a suitable detector. For a frequency extremely close to ω0, the phases of the amplitudes from the various electrons of a given pulse are still extremely close to those for radiation at ω0, and the total amplitude and intensity are still very close to those at ω0. That is, although the frequency spectrum is subject to 100% fluctuations, there is a correlation length, Γω, in frequency. The size of is the correlation length Γω can be estimated by noting that in going from frequency ω to ω+Γω, the phase difference of radiation from electrons that are the pulsewidth ∆t apart changes by about 180◦, so that the amplitude for radiation at ω + Γω is no longer well correlated to that at frequency ω. At once, we expect that