Response to “Comment on ‘Chaotic electron trajectories in an electromagnetic wiggler free-electron laser with ion-channel guiding’” [Phys. Plasmas 17, 093103 (2010)]

In Ref. 1, we have studied chaotic behavior of an electron motion in the combined electromagnetic wiggler and ion-channel electrostatic fields. The authors of comment² have claimed that, in Ref. 1, incorrect initial conditions have been used based on normalized electromagnetic wave frequency $ω−w$ and normalized mean axial velocity $βb$⁠. The initial conditions used in Ref. 1 to generate Poincaré surface-of-section plots are $ψ0$⁠, $ρ0$⁠, $P0ψ$⁠, and $P0ρ$⁠. The normalized electromagnetic wave frequency $ω−w$ and mean axial velocity $βb$ are parameters of system and are not initial conditions. Therefore, the initial conditions used in Ref. 1 are correct with no error which grows exponentially with time. Note that all parameters and variables used in this response have the same definition as in Ref. 1.

In Ref. 1, it has been assumed that the normalized mean axial velocity $βb$ is constant equal to 0.93 to determine only the self-magnetic field. The assumption of a constant mean axial velocity has been used in the previous studies of chaotic dynamic (see e.g., Refs. 3–5). The general form of dispersion relation, for an electromagnetic wiggler, can be written as $ω−w2=1-ω−b2βw/aw$⁠. Since the electron transverse velocity $βw$ is very complicated in a non-steady-state and chaotic system, the general form of dispersion relation is very complicated. So, we have used an approximate value for dispersion relation (i.e., $ω−w=1$⁠; the vacuum value of dispersion relation) in Ref. 1. For an electron beam with sufficiently low electron density (i.e., $nb=0$⁠), self-fields can be neglected [see Eqs. (4) and (5) of Ref. 1]. In this case, the normalized electron plasma frequency $ω−b[=(4πe2nb/mkw2c2)1/2]$ goes to zero and thus the dispersion relation reduces to its vacuum value (i.e., $ω−w=1$⁠). Therefore, the results presented in Ref. 1 for sufficiently low electron density or $ω−b=0$ (results of Figs. 1–3) are completely correct.

Now, we investigate the effects of more accurate values of mean axial velocity and dispersion relation on electron dynamic for the case of $ω−b≠0$ by using a self-consistent method to calculate simultaneously the mean axial velocity $βb=<βz>$⁠, mean value of electromagnetic wave frequency $<ω−w>$ and electron trajectories for Poincaré surface-of-section plots. Then we compare the results of this generalization with the results of Ref. 1.

Figure 1 shows Poincaré surface-of-section plots obtained by self-consistent method, for (a) $ω−b=0.5$ and (b) $ω−b=1.25$_, corresponding to Figs. 4(a) and 5 of Ref. 1, respectively. Other parameters are the same as for Figs. 4(a) and 5 of Ref. 1. Comparing Fig. 1(a) with Fig. 4(a) of Ref. 1, we see that the Poincaré surface-of-section plots shown in these figures are nearly the same. Also, a comparison between Figs. 1(b) and 5 of Ref. 1 shows that, although the axial mechanical momentum $pz$ for some trajectories changes due to the mean axial velocity and dispersion relation calculated by self-consistent method, the nonchaotic behavior of electron motion remains without change. Therefore, a very accurate method (self-consistent method) confirms the results presented in Ref. 1 for $ω−b≠0$⁠. Note that the mean values of axial velocity calculated by self-consistent method are equal to 0.807 and 0.951 for $ω−b=0.5$ and $ω−b=1.25$⁠, respectively. These values differ from the constant mean axial velocity used in Ref. 1 (i.e., $βb=0.93$⁠) but these differences do not have any considerable effect on chaotic dynamic as previously reported in Ref. 6. The mean values of electromagnetic wave frequency $<ω−w>$ calculated by self-consistent method are equal to 0.988 and 0.999 for $ω−b=0.5$ and $ω−b=1.25$⁠, respectively. An interesting result is that these values of $<ω−w>$ are very close to the vacuum value of dispersion relation (i.e., 1) used in Ref. 1.

In addition, the authors of comment have claimed that most of the results in Ref. 1 are invalid because of using different values of electron kinetic energy $γ$ for initial conditions in Figs. 2 and 3. They then argued that the variation of $γ$ is not permissive for initial conditions. In contrast with the authors of comment opinion, the variation of total energy is not permissive for initial conditions and during the electron motion because the electron total energy is a constant of motion as we have shown in Ref. 1. Note that, in the case of magnetostatic wiggler with axial magnetic field and in the absence of self-fields, the electron kinetic energy $γ$ is equal to the electron total energy and is a constant of motion (see, e.g., Fig. 3 of Ref. 3). For this case, but in the presence of self-fields, the electron kinetic energy $γ$ is not constant for initial conditions and during the electron motion (see, e.g., Figs. 6 and 7 of Ref. 3).

The steady-state transverse velocity presented in Eq. (1) of comment has been obtained under the assumption $γ=const$ (see Ref. 7). Because of large variations of $γ$ due to the electrostatic potential of ion-channel and electron beam, the assumption $γ=const$ in Eq. (1) of comment is not permissible. As a result, the graph of steady-state normalized axial velocity $β0z$ (denoted by $βb$ in comment) shown in Fig. 3 of comment is not correct. Note that $βb$ in Eq. (1) and Fig. 3 of comment is not the mean axial velocity, it is the steady-state axial velocity $β0z$⁠. Although the dispersion relation [Eq. (2)] of comment is a correct equation, the graph of steady-state dispersion relation shown in Fig. 2 of comment is not correct because of using incorrect steady-state transverse velocity $β0w$⁠. Also, the Poincaré surface-of-section plots shown in Fig. 1 of comment are not correct because of using incorrect values of mean axial velocity and dispersion relation.

The correct steady-state normalized axial velocity $β0z$ and dispersion relation $ω−0w$ can be calculated by solving the equations obtained by setting Eqs. (14), (15), (17), and (18) of Ref. 1 equal to zero (see Ref. 5) and using steady-state dispersion relation $ω−0w2=1-ω−b2β0w/aw$⁠. The result is shown in Fig. 2, for (a) steady-state normalized axial velocity $β0z$ and (b) steady-state electromagnetic wave frequency $ω−0w$⁠. Here the electron total energy $H−'$ is constant and equal to 3. Using Eqs. (1) and (2) of comment, the graphs of $β0z$ and $ω−0w$ based on electron kinetic energy $γ=const$ are also shown in this figure (dashed lines) for comparison. It is observed that the graphs of $β0z$ and $ω−0w$ with $γ=const$ differ significantly from the graphs of $β0z$ and $ω−0w$ with $H−'=const$ near the electron transverse velocity resonance. Note that for graphs of $γ=const$⁠, the electron total energy $H−'$ is not constant and varies with varying ion-channel frequency $ω−i$ to compensate variation of ion-channel electrostatic potential. The variation of electron total energy is physically impossible.

The comment includes a number of incorrect statements such as “The electron current density exists in the absence of self-fields….” This statement is in contradiction with Ampère’s law because, according to this law, the electron current density generates self-magnetic field.

In summary, we have shown that the initial conditions used in Ref. 1 are correct. The vacuum value of dispersion relation (i.e., $ω−w=1$⁠) used in Ref. 1 is exact in the absence of self-fields (⁠$ω−b=0$⁠) and, therefore, in the absence of self-fields, the results presented in Ref. 1 are completely correct. In the presence of self-fields (⁠$ω−b≠0$⁠), a self-consistent method has been used to calculate very accurate values of mean axial velocity and dispersion relation. It has been shown that the deviation of mean axial velocity calculated by self-consistent method from the constant mean axial velocity used in Ref. 1 does not have any considerable effect on electron chaotic dynamic and the mean value of dispersion relation is very close to the vacuum value of dispersion relation. Therefore, in the presence and absence of self-fields, the results presented in Ref. 1 are correct.