Plural interactions of space charge wave harmonics during the development of two-stream instability

Abstract

We construct a cubically nonlinear theory of plural interactions between harmonics of the growing space charge wave (SCW) during the development of the two-stream instability. It is shown that the SCW with a wide frequency spectrum is formed when the frequency of the first SCW harmonic is much lower than the critical frequency of the two-stream instability. Such SCW has part of the spectrum in which higher harmonics have higher amplitudes. We analyze the dynamics of the plural harmonic interactions of the growing SCW and define the saturation harmonic levels. We find the mechanisms of forming the multiharmonic SCW for the waves with frequencies lower than the critical frequency and for the waves with frequencies that exceed the critical frequency.

Space Charge Wave with Broad Frequency Spectrum Formation in Transit Section of Klystron-type Two-stream FEL with Helical Electron Beam

“…Two-velocity electron beam 5 is weakdensity-modulated as a result of the nonlinear parametric interaction of the electromagnetic wave The two-stream instability develops in electron beam 5 within transit section 6. Amplification maximum of the longitudinal electron waves in this case attains for the waves with some optimal frequency opt  (9) [1,8,[11][12][13]. The key point of the discussed design is that the modulation frequency 3  is much smaller than the optimal frequency opt  , at which SCW has maximal growth increment due to the two-stream instability.…”

“…It follows from (15) that plural three-wave parametric resonances occur in helical two-stream electron beam on frequencies cr    [1,12,13]. Plural threewave parametric resonance occurrence is coupled with the fact consequent from (15): for the amplifying wave the coupling between the wavenumber real part and frequency is linear…”

“…2 that for helical two-stream beams this domain expands with increasing beam input angle  . This means that the use of helical two-speed electron beams is preferable in multiharmonic FELs, the primary task of which is the formation of a powerful electromagnetic signal with a broad frequency spectrum [1,12,13].…”

“…In such modes the space charge wave (SCW) is excited on the frequency having maximal amplification due to the two-stream instability [1][2][3][4][5][6][7][8][9][10][11]. Lately, there are also works investigating multiharmonic operation modes of TSFELs [1,12,13]. In these studies, multiharmonic TSFELs capable of powerful ultrashort electromagnetic field cluster formation and able to create powerful multiharmonic signal with a wide frequency spectrum are investigated.…”

“…A multiharmonic SCW is the source of multiharmonic waves in TSFELs. Previously performed researches [1,12,13] showed that SCW with a broad frequency spectrum (tens and more harmonics) is excited in straight two-stream relativistic electron beams, if the frequency of the first harmonic of SCW that grows due to the two-stream instability is much less than the two-stream instability critical frequency. The excitement of such SCW is caused by two factors.…”