Comment on “Relativistic Raman backscattering theory with application in free-electron lasers with helical wiggler and axial magnetic field” [Phys. Plasmas 11, 4483 (2004)]

In a recent paper, Kordbacheh, Maraghechi, and Aghahosseini (Ref. 1) considered the relativistic Raman backscattering from an electron beam plasma with helical wiggler and axial magnetic field. In such a plasma, using the notations of Ref. 1, there are longitudinal waves $(ω2,k2)$ with the linear dispersion relation $ϵKMA=1−ωp2Φ∕ω22=0$ and transverse waves $(ω3,k3)$ with the linear dispersion relation $DKMA(ω3,k3)=k32c2−ω32+ωp2ω3∕(ωc+ω3)+Ψ=0$⁠. Let us here write their dispersion relation [Eq. (36) in Ref. 1], which accounts for the wiggler induced coupling between the longitudinal and transverse waves, in the form

where

However, the Raman backscattering of circularly polarized electromagnetic waves in a magnetized plasma² described by a macroscopic model was previously reconsidered by means of a fully relativistic treatment which is valid for arbitrary pump wave amplitudes.³ The dispersion relation was later generalized to include also kinetic effects.⁴ The general dispersion relation was then written in the form

The explicit expressions for the dielectric constant $ϵ=ϵ(ω,k)$ as well as for the dispersion functions $D±=D(ω∓ω0,k∓k0)$ and the parameters $a±$ and $b$ can be found in Ref. 4. Here $(ω,k)$ represents the longitudinal wave, $(ω0,k0)$ the pump wave, and $B⊥0$ the magnetic wiggler field amplitude. We note that Eq. (2) has been derived for arbitrarily large pump wave amplitudes. As some readers may think that it is approximate as its right-hand side contains second-order terms, it should be stressed that all higher order terms have been fully included in its derivation, and then collected in Eq. (2) above. Thus, we stress again that Eq. (2) is valid for any pump field strengths. This is in contrast to Eq. (1) above which obviously is not correct.

Equation (2) shows how longitudinal waves with the linear dispersion relation $ϵ=0$⁠, couple with transverse waves with the dispersion relations $D+=0$ and $D−=0$⁠. For three wave interactions, assuming that $(ω,k)$ and $(ω−ω0,k−k0)$ are resonant modes, we can approximate Eq. (2) by

By comparing the equations above we find some very significant differences. The right-hand side of Eq. (3) is obviously always negative, whereas this is not always the case for the right-hand side of Eq. (1) if $a1≠0$⁠. Thus, the spatial growth rate $ΔL$ differs widely from those of the results for medium values of the axial magnetic field when the wiggler induced electron quiver velocity is comparable to the axial velocity. In addition, the squared growth rate $ΔL2$ is becauseof Eq. (3) positive, whereas in Ref. 1 it can be negative in certain frequency intervals.

What is even more important is that neither Eq. (1) nor Eq. (3) should be used when the wiggler induced electron velocity is comparable to the axial velocity. In that case, the numerical calculations in Ref. 1 ought to have been based on Eq. (2), instead of Eqs. (1) and (3). The figures of Ref. 1 are therefore not reliable.

We conclude that for the strong wiggler fields considered in Ref. 1, it is necessary to use a four-wave interaction theory [leading to Eq. (2)] instead of a three-wave interaction theory [leading to Eqs. (3) and (1)].