⇦Chinese Physics Letters, 2017, Vol. 34, No. 7, Article code 075202 Forming of Space Charge Wave with Broad Frequency Spectrum in Helical Relativistic Two-Stream Electron Beams * A. Lysenko1**, I. Volk1, A. Serozhko1, O. Rybalko2 Affiliations 1Department of Applied Mathematics and Complex Systems Modelling, Sumy State University, Sumy 40007, Ukraine 2Department of Electrical Engineering, Technical University of Denmark, Kgs. Lyngby 2800, Denmark Received 14 March 2017 *Supported by the Ministry of Education and Science of Ukraine under Grant No 0117U002253.
**Corresponding author. Email: lysenko_@ukr.net Citation Text: Lysenko A, Volk I, Serozhko A and Rybalko O 2017 Chin. Phys. Lett. 34 075202 Abstract We elaborate a quadratic nonlinear theory of plural interactions of growing space charge wave (SCW) harmonics during the development of the two-stream instability in helical relativistic electron beams. It is found that in helical two-stream electron beams the growth rate of the two-stream instability increases with the beam entrance angle. An SCW with the broad frequency spectrum, in which higher harmonics have higher amplitudes, forms when the frequency of the first SCW harmonic is much less than the critical frequency of the two-stream instability. For helical electron beams the spectrum expands with the increase of the beam entrance angle. Moreover, we obtain that utilizing helical electron beams in multiharmonic two-stream superheterodyne free-electron lasers leads to the improvement of their amplification characteristics, the frequency spectrum broadening in multiharmonic signal generation mode, and the reduction of the overall system dimensions. DOI:10.1088/0256-307X/34/7/075202 PACS:52.35.-g, 52.35.Mw © 2017 Chinese Physics Society Article Text The great amount of attention paid to two-stream instability research is caused mainly by its applicability to two-stream free electron lasers (FELs).[1-20] Due to high growth rates of two-stream instability, two-stream FELs are characterized by exceptionally high amplification rates.[1,21,22] One of the two-stream instability features is the excitation of the large number (dozens, hundreds) of space charge harmonics during its development.[1,9,23-25] Forming of space charge wave (SCW) with the broad frequency spectrum, in which higher harmonics have higher amplitudes, becomes possible. Forming of such SCWs gives the ability to realize multiharmonic operation modes in two-stream FELs and to form ultrashort wavepackets of electromagnetic waves. Therefore, the study of multiharmonic properties of two-stream instability and the investigation of regimes which allow effective forming of the SCW with wide frequency spectrum are very important and actual tasks. SCW multiharmonic property investigation in straight relativistic two-stream electron beams is described in earlier works.[1,9,23-25] We have shown that if the frequency of the first SCW harmonic in the two-stream electron beam is much less than the critical frequency of two-stream instability, the multi-harmonic SCW, which consists of tens or more of harmonics with comparable amplitudes, will be excited in such a system.[1,9,23-25] Such SCW excitation is caused by the quasilinear dispersion characteristic of this SCW. Therefore, the satisfaction of three-wave parametric resonance conditions is possible between the harmonics of such a wave. As a result, the harmonics are amplified due to both the two-stream instability and the plurality of three-wave parametric resonances. Thus general dynamics of wave harmonics in the two-stream electron beam turns out to be complex enough. The present research is devoted to the multiharmonic processes investigation in helical relativistic two-stream electron beams. The analysis carried out in this study shows helical two-stream electron beam allowance for more effective forming of SCWs with broad frequency spectrum compared with straight beams. We consider the following model. The helical beam contains two partial interpenetrating electron streams with numerically close partial relativistic velocities $\upsilon _1$, $\upsilon _2$ ($\upsilon _1 -\upsilon _2\ll\upsilon _1,\upsilon _2$). It moves at the angle $\alpha$ to the focusing magnetic field ${\boldsymbol B}_0$ (Fig. 1) along the helical trajectory. Plasma frequencies of partial electron beams are considered to be equal, i.e., $\omega _{\rm p1} =\omega _{\rm p2} =\omega _{\rm p}$, and the beam space charge is compensated for by the ion background.
cpl-34-7-075202-fig1.png
Fig. 1. The relativistic helical two-stream electron beam model.
We restrict ourselves by considering the model with transversal dimensions much less than the wavelengths of waves propagating in the investigated system. In this case the beam can be considered homogenous in the transverse plane and the influence of its boundaries on the wave dynamics can be neglected. We consider the SCW electric field defining the dynamics of two-stream instability as multiharmonic $$\begin{align} E_{\rm z} =\sum\limits_{m=1}^N[E_m \exp (ip_m)+{\rm c.c.}] ,~~ \tag {1} \end{align} $$ where $N$ is the number of harmonics considered, $p_m =\omega _m t-k_m z$ is the phase, $\omega _m =m\cdot \omega _1$ is the frequency of the $m$th SCW harmonic, $k_m$ is its wavenumber, and ${\boldsymbol e}_z$ is the unit vector of axis $Z$ with axis $Z$ being directed along the focusing magnetic field ${\boldsymbol B}_0$. As an initial step we use a quasihydrodynamic equation,[1,21,22] the continuity equation and Maxwell's equations. We use the hierarchic approach to the theory of oscillations and waves.[1,23] This approach utilizes the Krylov–Bogolyubov averaging method[1,26] for the asymptotic integration of differential equations. As a result, we derive a system of $m$ differential equations for complex amplitudes of SCW electric field strength harmonics in the small-signal approximation $$\begin{alignat}{1} \!\!\!\!\!\!&C_{2,m} \frac{d^2E_m}{dz^2}+C_{1,m} \frac{dE_m}{dz}+D_m E_m \\ \!\!\!\!\!\!=\,&C_{3,m} \langle E_z \cdot \sum\limits_{{m}'=1}^N [E_{{m}'} \exp (ip_{{m}'})/({i{m}'})+{\rm c.c.}] \rangle _{p_m},~~ \tag {2} \end{alignat} $$ where $$\begin{alignat}{1} \!\!\!\!\!\!D_m (\omega _m,k_m)\equiv -ik_m \Big(1-\sum\limits_{q=1}^2 \frac{\omega _{\rm p}^2 (1-(\upsilon _{qz} /c)^2)}{(\omega _m -k_m \upsilon _{qz})^2\gamma _q}\Big),~~ \tag {3} \end{alignat} $$ is the dispersion function of SCW, $\upsilon _{qz}$, $\gamma _q =(1-(\upsilon _q /c)^2)^{-1/2}$ are the $Z$ component of the velocity and the relativistic factor of $q$th electron beam respectively, $c$ is the light velocity, $C_{1,m} =\partial D_m /\partial (-ik_m)$, $C_{2,m} =\partial ^2D_m /\partial (-ik_m)^2/2$, $\langle\ldots\rangle_{p_m} =\frac{1}{2\pi}\int_0^{2\pi} {(\ldots\cdot \exp (-ip_m))dp_m}$, $$\begin{align} C_{3,m} =\,&\sum\limits_{q=1}^2 \frac{3e\omega _{\rm p}^2 (1-(\upsilon _{qz} /c)^2)k_1}{im(\omega _1 -k_1 \upsilon _{qz})^3\upsilon _{qz} \gamma _q^{\rm 2} m_e}\\ &\cdot\Big[\frac{\omega _1 (1-(\upsilon _{qz} /c)^2)}{\omega _1 -k_1 \upsilon _{qz}}-1\Big], \end{align} $$ and $e$ and $m_e$ are the electron charge and mass, respectively. Note that the dispersion equation $D_m (\omega _m,k_m)=0$ has complex solutions in the two-stream instability realization case. At that, real components of frequencies and wavenumbers are used in the phases of harmonics. Therefore, dispersion function (3) has a nonzero value when we substitute the real components of frequencies and wavenumbers into Eq. (3). This fact is considered by nonzero component $D_m E_m \ne 0$ in Eq. (2). It is also worth noting that we should consider the second derivative component in Eq. (2) to describe the two-stream instability dynamics. This component defines the SCW growth rate. Thus in the case of an absence of parametric resonances between SCW harmonics ($C_{3,m} \langle {E_z \cdot \sum\limits_{{m}'=1}^N {[{E_{{m}'} \exp (ip_{{m}'})/({i{m}'})+{\rm c.c.}}]}} \rangle _{p_m} =0$), the SCW growth rate is approximately defined by expression $(-D_m /C_{2,m})^{1/2}$. It is considered here that $C_{2,m} \cdot d^2E_m /dz^2, D_m E_m \gg C_{1,m} \cdot dE_m /dz$. We take the system parameters to observe the two-stream instability.[1,21,22] In that case the dispersion equation for the SCW in a helical two-stream electron beam, $$\begin{align} D_m (\omega _m,k_m)\equiv\,&-ik_m \Big({1-\sum\limits_{q=1}^2 {\frac{\omega _{\rm p}^2 (1-(\upsilon _{qz} /c)^2)}{(\omega _m -k_m \upsilon _{qz})^2\gamma _q}}}\Big)\\ =\,&0,~~ \tag {4} \end{align} $$ has complex solutions. An approximate analytical solution to Eq. (4) for a helical relativistic two-stream electron beam can be found in the same way as in the straight beam case (e.g., Refs. [1,22]). We find the solution in the following way, $$\begin{align} k_m =\omega _m /\upsilon _{0z} +i{\it \Gamma} _m,~~ \tag {5} \end{align} $$ where $\upsilon _{0z} =(\upsilon _{1z} +\upsilon _{2z})/2$, and $i{\it \Gamma} _m$ is the nonlinear addition to the SCW wavenumber. We substitute Eq. (5) into Eq. (4) and after simple algebraic manipulations we obtain an equation for the nonlinear addition to the SCW wavenumber, $$\begin{align} i{\it \Gamma} _m =\,&\pm \frac{\omega _p}{\upsilon _{0z}}\sqrt {\frac{1-(\upsilon _{0z} /c)^2}{\gamma _0}}\\ &\cdot \Big\{1+\frac{\omega _m^2 \delta ^2\gamma _0}{\omega _p^2 ( {1-(\upsilon _{0z} /c)^2})}\\ &\pm \sqrt {1+\frac{4\omega _m^2 \delta ^2\gamma _0}{\omega _p^2 ({1-(\upsilon _{0z} /c)^2})}} \Big\}^{1/2},~~ \tag {6} \end{align} $$ where $\gamma _0 =1/\sqrt {1-(\upsilon _0 /c)^2}$, $\upsilon _0 =(\upsilon _1 +\upsilon _2)/2$ is the two-stream electron beam average velocity, $$\begin{alignat}{1} \!\!\!\!\!\!\delta =\frac{\upsilon _{z1} -\upsilon _{z2}}{\upsilon _{z1} +\upsilon _{z2}}=\frac{\upsilon _1 \cos \alpha -\upsilon _2 \cos \alpha}{\upsilon _1 \cos \alpha +\upsilon _2 \cos \alpha}=\frac{\upsilon _1 -\upsilon _2}{\upsilon _1 +\upsilon _2}.~~ \tag {7} \end{alignat} $$ Equation (6) is derived under conditions $| \upsilon _{0z} {\it \Gamma} _m /\omega _p | \ll 1$ and $| \omega _m \delta /\omega _p |\ll 1$. From Eq. (6), in the case that the SCW harmonic frequency is less than the critical frequency of helical two-stream electron beam, $$\begin{align} \omega _{\rm cr} =\frac{\sqrt 2 \omega _p}{\delta}\sqrt {\frac{1-(\upsilon _{0z} /c)^2}{\gamma _0}},~~ \tag {8} \end{align} $$ there are two waves characterized by complex wavenumber $k$. This means that the occurrence of the convective instability is possible in the investigated system.[27] The complex nature of the wavenumber is a necessary but not sufficient condition. According to Ref. [27] the sufficient condition is formulated as follows: to decide whether a given wave with a complex $k={\rm Re}(k)+i{\rm Im}(k)$ for some real $\omega$ is amplifying or evanescent, determine whether or not ${\rm Im}(k)$ has a different sign when the frequency takes on a large negative imaginary part. If it does, then the wave will be amplifying, otherwise it will be an evanescent wave. Let us check the above mentioned criteria for the helical two-velocity relativistic electron streams when $\omega < \omega _{\rm cr}$. For this purpose we calculate the imaginary part of the wavenumber for the frequencies $\omega _1 =\omega _{\rm cr} \sqrt {3/8}$ and $\omega _2 =\omega _{\rm cr} \sqrt {3/8} \cdot (1-i)$. Using Eq. (6) for the frequency $\omega _1$ we find that $i{\rm Im}(k_1)=i{\it \Gamma} _{1} =+i0.5({\omega _p /\upsilon _{0z}})\sqrt {({1-(\upsilon _{0z} /c)^2})/\gamma _0}$. For the complex frequency $\omega _2$ at the same signs in Eq. (6) we achieve $i{\rm Im}(k_2)\approx -i0.61({\omega _p /\upsilon _{0z}})\sqrt {( {1-(\upsilon _{0z} /c)^2})/\gamma _0}$. Therefore, the sign of the wavenumber imaginary part changes when the frequency takes on a large enough negative imaginary part. This means that the criterion stated above is satisfied and the convective instability realizes in helical two-stream relativistic electron beams when $\omega < \omega _{\rm cr}$. Therefore, in the investigated system one of the waves characterized by a complex wavenumber grows exponentially (growing wave). For this wave the quantity ${\it \Gamma} _m$ has the physical meaning of the growth rate. The other wave attenuates exponentially (attenuated wave). Since the attenuated wave amplitude decreases rapidly, we do not take this wave into further consideration. One should also note that Eq. (4) assumes two real solutions despite two complex solutions when $\omega < \omega _{\rm cr}$. These real solutions correspond to slow and fast waves.[1,21,22] We assume that upon entering the considered system amplitudes of the slow and the fast SCWs are small. Therefore, the influence of these waves on the two-stream instability development processes can be neglected. From the analysis of Eq. (6) the optimal frequency for the helical two-stream electron beam can be found, $$\begin{align} \omega _{\rm opt} =\frac{\sqrt 3 \omega _p}{2\delta}\sqrt {\frac{1-(\upsilon _{0z} /c)^2}{\gamma _0}} =\omega _{\rm cr} \sqrt {\frac{3}{8}}.~~ \tag {9} \end{align} $$ It corresponds to the maximal growth rate $$\begin{align} {\it \Gamma} (\omega _{\rm opt})=\frac{\omega _p}{2\upsilon _{0z}}\sqrt {\frac{1-(\upsilon _{0z} /c)^2}{\gamma _0}}.~~ \tag {10} \end{align} $$ From an analysis of Eqs. (8)-(10) we can make sure that with increasing the entrance angle $\alpha$, both the critical frequency $$\begin{align} \omega _{\rm cr} =\frac{\sqrt 2 \omega _p \sqrt {1+\gamma _0^2 (\upsilon _0 /c)^2\sin ^2\alpha}}{\delta \cdot \gamma _0^{3/2}},~~ \tag {11} \end{align} $$ and the maximal growth rate $$\begin{align} {\it \Gamma} (\omega _{\rm opt})=\frac{\omega _p \sqrt {1+\gamma _0^2 (\upsilon _0 /c)^2\sin ^2\alpha}}{2\upsilon _0 \gamma _0^{3/2} \cos \alpha},~~ \tag {12} \end{align} $$ increase. Therefore, in helical two-stream electron beams the two-stream instability development occurs with higher growth rates than in straight beams. That is why utilizing of helical beams in two-stream superheterodyne free-electron lasers leads to the increase of electromagnetic wave amplification rates. Note that the same conclusion has been carried out for the two-stream FEL with helical beams,[19,20] but the reason for this increase has not been defined. As follows of the foregoing analysis, the electromagnetic signal amplification rates increasing in the two-stream SFELs relates to the increase of the two-stream instability growth rates in helical relativistic electron beams. We should also mention that for helical electron beams both optimal and critical frequencies of the two-stream instability are higher compared with straight electron beams. This means that two-stream SFELs with helical electron beams can operate on higher frequencies compared with SFELs utilizing straight beams. From Eq. (5) we can realize that plural three-wave parametric resonances take place in a helical two-stream electron beam on frequencies $\omega < \omega _{\rm cr}$.[1,9,23-25] An occurrence of the plural parametric resonances relates to the fact consequent from Eq. (6), and for the growing wave the connection between the real part of the wavenumber and frequency is linear, $$\begin{align} {\rm Re}(k)=\omega /\upsilon _{0z},~~ \tag {13} \end{align} $$ which means that if the frequency of the $m$th harmonic, $\omega _m$, is $m$ times greater than the frequency of the first harmonic $\omega _1$, then the real part of the wavenumber of this harmonic ${\rm Re}(k_m)$ will be $m$ times greater than the real part of the wavenumber of the first harmonic ${\rm Re}(k_1)$ (i.e., $\omega _m < \omega _{\rm cr}$), $$ {\rm Re}(k_m)=\omega _m /\upsilon _{0z} =m\omega _1 /\upsilon _{0z} =m{\rm Re}(k_1). $$ Therefore, the phase of the $m$th harmonic (defined by the real part of the wavenumber) $$\begin{align} p_m =\,&\omega _m t-{\rm Re}(k_m)z\\ =\,&m\cdot \omega _1 t-m\cdot {\rm Re}(k_1)z=m\cdot p_1~~ \tag {14} \end{align} $$ surpasses the first harmonic phase $m$ times. This leads to the parametric resonance conditions fulfilling in the two-stream system for the plurality of SCW harmonics meeting the requirement $\omega < \omega _{\rm cr}$, $$ p_{m_1} =p_{m_2} +p_{m_3}, $$ or according to Eq. (14) $$\begin{align} m_1 =m_2 +m_3 ,~~ \tag {15} \end{align} $$ where $m_1$, $m_2$ and $m_3$ are the whole numbers. Condition (15) realizes with the participation of a great number of harmonics, e.g., $3=2+1$, $3=5-2$, $3=7-4$ and $6=3+3$. Therefore, we are talking about this situation as being about that sort in which the plural parametric resonances are realized. These plural resonant interactions are taken into account in the system (2) in $C_{3,m} \langle E_z \cdot \sum\limits_{{m}'=1}^N [{E_{{m}'} \exp (ip_{{m}'})/({i{m}'})+c.c.}]\rangle_{p_m}$ component. Thus the plural parametric resonances between SCW harmonics realized in the investigated system are due to the linear dispersion dependence while $\omega < \omega _{\rm cr}$. Because of plural three-wave parametric resonant interactions, the forming of SCW with a broad multiharmonic frequency spectrum occurs. In this spectrum higher harmonics have higher amplitudes than lower harmonics. The feature of the SCW with broad frequency spectrum forming in helical electron beams can be described as follows: according to Eq. (11) the critical frequency for the helical beams increases with the electron beam input angle $\alpha$ to the longitudinal focusing magnetic field. This means the increase of the frequency domain, in which the forming of a multiharmonic spectrum occurs. This fact is illustrated in Fig. 2, which shows the dependences of the two-stream instability growth rates on frequency for different input angles $\alpha$. These dependences are derived from the numerical solution to Eq. (4).
cpl-34-7-075202-fig2.png
Fig. 2. Two-stream instability growth rates with different input angles $\alpha$.
We consider the case of $\omega _{p1} =\omega _{p2} =\omega _p =1.5\cdot 10^{11}$ s$^{-1}$, $\gamma _1 =4.8$ and $\gamma _2 =4.2$. Curve 1 corresponds to $\alpha =0^{\circ}$, curve 2 corresponds to $\alpha =10^{\circ}$, curve 3 corresponds to $\alpha =20^{\circ}$, and curve 4 corresponds to $\alpha =30^{\circ}$. The first harmonic frequency is $\omega _1 =0.6\cdot 10^{12}$ s$^{-1}$. The conclusions made from the analysis of Eqs. (11) and (12) follow Fig. 2: with the increase of the two-stream beam input angle both the growth rate ${\it \Gamma}$ and the critical frequency $\omega _{\rm cr}$ increase. When the first SCW harmonic frequency $\omega _1$ is much less than the critical frequency, the plural parametric resonances between SCW harmonics occur in the frequency domain $\omega _1 < \omega _m < \omega _{\rm cr}$. By analyzing Fig. 2, we can see that for helical two-stream beams this domain increases with the beam input angle $\alpha$. This means that utilizing of helical two-stream electron beams is preferable in the multiharmonic FELs, where the primary task is the forming of a powerful electromagnetic signal with the broad frequency spectrum.[1,9,23] Figure 3 represents the dependence of the electric field strength amplitudes of ten SCW harmonics on longitudinal coordinate $z$ for the beam input angles $\alpha =0^{\circ}$ (curves 1) and $\alpha =20^{\circ}$ (curves 2). Calculation parameters are the same as those in Fig. 2. In both cases the SCW on the system input is monochromatic with the frequency $\omega _1 =0.6\cdot 10^{12}$ s$^{-1}$, i.e., consists of one harmonic. The calculation of the dependences is performed by means of the equation system for the electric field strength harmonic amplitudes of Eq. (2). Based on the results presented in Fig. 3, the electric field strength harmonics amplification rates for the helical electron beam case (curves 2) are higher than the straight electron beam case (curves 1). This confirms the analysis carried out above. Due to the plural three-wave parametric resonances the higher harmonics are excited and then they are amplified because of the two-stream instability also confirmed by Fig. 3. Because of the processes the SCW with the broad frequency spectrum is formed.
cpl-34-7-075202-fig3.png
Fig. 3. Dependences of the electric field strength amplitudes of ten SCW harmonics on longitudinal coordinate $z$ for the beam input angles $\alpha =0^{\circ}$ (curves 1) and $\alpha =20^{\circ}$ (curves 2).
We should notice that with increasing the harmonic number, their growth rates increase up to the optimal frequency (Fig. 2), thus the amplification rate of the higher harmonics turns out to be higher. There is an abnormal spectrum part in which higher harmonics have higher amplitudes to form. It must also be noted that the equation system (2) defining Fig. 3 is achieved in the small signal approximation. This means that Fig. 3 does not describe saturation processes. Nevertheless, based on Fig. 3 we can obtain that two-stream SFELs utilizing helical electron beams have smaller longitudinal dimensions compared with the two-stream SFELs utilizing straight beams. In summary, we have constructed a quadratic nonlinear theory of plural interactions of growing space charge wave harmonics during the development of the two-stream instability in helical relativistic electron beams. We have found that the two-stream instability growth rate increases with the beam input angle. This means that two-stream superheterodyne FELs utilizing the two-stream instability in helical electron beams as an additional amplification mechanism have higher amplification rates compared with TSFELs utilizing straight electron beams. One should also expect that in this kind of device the saturation occurs earlier. It means that such FELs can have smaller overall dimensions. We have demonstrated that due to the linearity of the growing SCW dispersion characteristic the parametric resonance conditions for helical two-stream electron beam are satisfied for all harmonics, where the frequency is less than the critical frequency of the two-stream instability. The simultaneous excitation of many harmonics occurs as a result of the plurality of the three-wave parametric resonances. These harmonics grow due to both the parametric interactions and two-stream instability. Thus the modes of operation, in which the excitation of dozens of harmonics with comparable amplitudes and SCW with broad frequency spectrum forming occur, become possible. We have found out that the width of such a frequency spectrum increases with the helical two-stream electron beam input angle $\alpha$. This fact is associated with the increase of critical frequency $\omega _{\rm cr}$ for helical electron beams compared with straight beams and has a relativistic nature. Moreover, we have obtained that utilizing of helical electron beams in multiharmonic two-stream superheterodyne free-electron lasers leads to the improvement of their amplification characteristics, the frequency spectrum broadening in multiharmonic signal generation mode, and the reduction of the system overall dimensions. References Two‐stream, free‐electron lasersGain enhancement in a free electron laser by two?stream instabilityDouble beam free electron laserDouble?stream cyclotron maserGrowth and saturation of stimulated beam modulation in a two‐stream relativistic klystron amplifierNonlinear self-consistent theory of two-stream superheterodyne free electon lasersTwo-Stream Free Electron Lasers: Physical Analysis of the Systems with Monochromatic PumpingTwo-beam free-electron laserInstability of Two-stream Free-electron Laser with an Axial Guiding Magnetic FieldDispersion relation and growth in a two-stream free electron laser with helical wiggler and ion channel guidingThe effects of self-fields on the electron trajectory and gain in a two-stream electromagnetically pumped free-electron laser with axial guiding fieldThree-dimensional simulation of harmonic up-conversion in a prebunched two-beam free-electron laserEffects of finite beam and plasma temperature on the growth rate of a two-stream free electron laser with background plasmaEffects of self-fields on electron trajectory and gain in planar wiggler free-electron lasers with two-stream and ion-channel guidingInstability of wave modes in a two-stream free-electron laser with a helical wiggler and an axial magnetic fieldWave mode instabilities in a two-stream free-electron laser with a background plasmaEffect of two-stream instability on the saturation mechanism of a two-stream free-electron laser with a helical wiggler pumpSaturation mechanism in a two-stream free-electron laserHierarchical Asymptotic Methods in the Theory of Cluster Free Electron LasersEffect of parametric resonance on the formation of waves with a broad multiharmonic spectrum during the development of two-stream instabilityPlural interactions of space charge wave harmonics during the development of two-stream instability
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