Spectral limits and frequency sum-rule of current and radiation noise measurement

The current noise spectrum of an electron beam is generally considered white and expressed by the shot-noise formula (eI₀). It is possible to control the spectral energy of a random electron beam current by longitudinal space charge micro-dynamics and dispersive transport. Both noise suppression (relative to eI₀) and noise enhancement have been demonstrated, exhibiting sub/super-Poissonian particle distribution statistics, respectively. We present a general theory for the current noise of an e-beam and its radiation emission in the entire spectrum. The measurable current noise spectrum is not white. It is cut-off at high frequencies, limited by the measurement length and the beam axial momentum spread (fundamentally limited by quantum uncertainty). We show that under certain conditions the current noise spectrum satisfies a frequency sum-rule: exhibiting noise enhancement in one part of the spectrum when suppressed at another part and vice versa. The spontaneous emission (radiation noise) into a single radiation mode or single direction in any scheme (OTR, Undulator etc.) is sub-radiant when the beam current is sub-Poissonian and vice versa, but the sum-rule does not apply.