1.Introduction

The idea to use the two-stream instability to amplify the electromagnetic waves appeared long after its theoretical discovery[1,2]and experimental con firmation[3,4].A vast number of articles,reviews and books are devoted to this question nowadays,e.g.[5,6].The main advantage of the devices based on this phenomenon lies in the ability to increase their working frequencies.In particular,it was shown that the working frequency of such devices depends inversely on the difference between partial velocitiesBy choosing the velocities of the electron beams close enough,we achieve the principal ability to increase the working frequencies abruptly.However,non-relativistic systems did not allow the realization of this ability to the full extent.In experiment,Δυ could not be lower than the value of the electron velocity thermal spread.In this case,the two-velocity beam became a one-stream one.

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Interest in the two-stream electron-wave systems was renewed with the appearance of devices utilizing relativistic electron beams—the two-stream superheterodyne free-electron lasers(FELs)[7–28].With the increase of the electron beam energy in conditions of fixed energy distribution,the electron velocity distribution decreases.This fact was also facilitated by the progress in the techniques of the formation of high-current relativistic electron beams[29].The described devices achieved an ability to operate in submillimeter infrared wavebands.

It was also found that two-stream superheterodyne FELs[7–28]are characterized by exceptionally high rates of electromagnetic wave amplification due to the two-stream instability acting as an additional amplification mechanism[5–7,30,31].Most works[8–24]devoted to the two-stream superheterodyne FELs focus on the regimes in which maximal amplification of the signal occurs.In these models,the frequency of the first space-charge wave(SCW)harmonic coincide with the frequency(optimal frequency)at which maximal SCW growth due to the two-stream instability takes place.

Significant attention both in science and technology is now being paid to the field of creating and researching the systems able to form powerful ultrashort electromagnetic pulses(including femtosecond ones),signals with a broad frequency spectrum[7,32,33].Such signals have wide practical application in a series of fundamental and applied researches in physics,chemistry,biology and medicine[32,33].As shown by different research,two-stream superheterodyne FELs can operate in a powerful ultrashort electromagnetic cluster formation mode[7,25–28].In this case,the frequency of the first harmonic must be much lower than both optimal and critical frequencies of the two-stream instability[7,30,31].The main purpose of such FELs is not the maximal signal amplification but the forming of electromagnetic waves with a broad frequency spectrum.These devices can be spoken of as multiharmonic FELs[7,25–28].

In multiharmonic two-stream superheterodyne FELs the SCW excitation in the two-stream relativistic electron beam acts as the source of multiharmonic waves.The excitation of the SCW with a broad frequency spectrum is determined by the following.First,the SCW growth due to the two-stream instability is characterized by the quasilinear dispersion dependence of the instability[5–7,30,31].Therefore,if the first SCW harmonic frequency is much lower than the critical frequency of the two-stream instability ωcr,then three-wave parametric resonance conditions are fulfilled for the plurality of SCW harmonics,with frequencies lower than ωcr.In this connection,the higher SCW harmonics are excited in the two stream system due to the plural three-wave resonant interactions[7,25–28,34–36].Second,as long as the first harmonic of the SCW is much lower than the two-stream instability critical frequency,all the harmonics excited due to the plural resonant interactions are also amplified due to the two-stream instability.As a result of the superposition of plural three wave resonant interactions and the two-stream instability,the multiharmonic SCW with a broad frequency spectrum forms.This wave acts as a source of multiharmonic waves in multiharmonic two-stream superheterodyne FELs.

Nowadays,the dynamics of multiharmonic SCW harmonics in straight two-stream relativistic electron beams(REBs)is investigated in[34,35].In these works,saturation levels and lengths were described along with the multiharmonic SCW frequency spectrum width.In[36],helical two-stream REBs are investigated.It is concluded that helical two-stream REBs are characterized by higher growth rates and higher two-stream instability critical frequencies in comparison with straight beams.Due to this fact,they can form SCWs with wider frequency spectra and saturation lengths ought to be smaller.However,saturation levels and spectra of such waves were not found in a cubic non-linear approximation.There is also the question of how saturation levels and frequency spectrum widths depend on different two-stream REB parameters.We considered the influence of the relativistic factors of partial beams,the relation between longitudinal and transversal velocities of helical electron beams,and the electron concentration of two-stream beams.The solution to this question is the aim of the present article.

Figure 1.Model of the helical two-stream REB.

2.Model

In general,we consider the helical two-stream relativistic electron beam consisting of two partial interpenetrative electron streams.The streams have close values of partial relativistic velocities υ1,υ2and equal partial plasma frequencies ωp,1　=　ωp,2　=　ωp.Such a beam enters the system at an angle α to the focusing magnetic field B0( figure 1)and propagates along the helical trajectory.We consider the beam as homogeneous in the transverse plane and neglect thermal velocity distribution and electron collisions.We assume that an ion background compensates the beam space charge.

Here,we consider the case where the two-stream instability in the two-stream REB takes place.This means that in the investigated system the SCW with the amplitude growing exponentially dominates over other types of waves[5–7,30,31].In general,we consider such a wave as multiharmonic,where the electric field strength has the form

is also m times greater than the phase of the first harmonic.As a result,three-wave parametric resonance conditions are fulfilled for the plurality of SCW harmonics with frequencies ωm ＜ ωcr

If the frequency of the first SCW harmonic ω1is much lower than the critical frequency ωcr,than all the SCW harmonics,with frequencies lower than critical(ωm ＜ ωcr)are amplified due to the two-stream instability.Despite this,plural three-wave parametric resonances occur in the helical two-stream electron beam at frequencies ωm ＜ ωcr[7,25–28,34–36].The reason for the occurrence of plural parametric resonances is related to the linearity of the bond between the real part of the wavenumber and the frequency in the case of a growing wave[5–7,30,31]

In order to solve the motion problem,we utilize the characteristic of equation(6)[7,27,39].This equation is an ordinary differential equation.Since we consider the boundary problem,we move from the time derivative to the coordinate derivative using the well-known relation for the velocityWe supplement the equation system by the equations for fast phases pq,m.We consider that amplitudes of fields change slowly with the change of longitudinal coordinate z.The slow longitudinal coordinateζ=zξ is used to describe slow amplitude changes.As a result,we get the equation system in standard form

is m times greater than the wavenumber of the first harmonic.Hence,the phase of the mth harmonic

where N is the maximal harmonic number taken into account in the simulations;　pm　　=ω　m　t　-km 　zis the mth harmonic phase,ω　m=m⋅ω1,kmare its frequency and wavenumber,respectively;the direction of the Z axis coincides with the direction of the focusing magnetic field B0( figure 1).

or

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where m1,m2,m3are integers.Condition(5)covers many harmonics.Therefore,at ωm ＜ ωcrplural three-wave parametric resonant interactions take place.As a matter of fact,we deal with the superposition of interacting harmonics that are also amplified due to the two-stream instability.

3.Field equations

We solve the motion problem and the continuity equation by means of the modernized method of averaged characteristics[7,39].To solve the electromagnetic field excitation problem,we utilize the method of slowly varying amplitudes.The features of plural three-wave parametric resonant interactions of SCWs and the electromagnetic signal are considered during the solution.

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We consider the case where the electron collision and thermal spread can be neglected.We investigate the model in which all values depend only on longitudinal coordinate z and time t.Then,the relativistic quasihydrodynamic equation,the continuity equation and Maxwell’s equations can be presented in the following form:

Here,are velocity projection on the Z axis and relativistic factor of the qth beam,respectively,c is the speed of light,meare the charge and the mass of electrons,respectively,nqis the electron concentration of the qth beam,Ezis the electric field strength given by(1).

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As is known,the solution of equations(6)–(8)consists of three formally independent stages:solving the motion problem(6)of the two-stream REB in given electromagnetic fields;solving the continuity equation(7),considering that we already know the velocities of the electron beam;solving the excitation problem of electromagnetic fields(8),considering the velocities and concentrations as known values.

In order to numerically analyze the processes described above,we initially use the quasihydrodynamic equation,Maxwell’s equations and the continuity equation.We utilize methods of hierarchic theory of oscillations and waves to solve these equations[7,27,28].This approach is based on the　Krylov–Bogoliubov　asymptotic　integration　method[7,37,38].

This means that the wavenumber of the mth harmonic

We compare system(9)–(11)with the standard[7,27 39]and write down the vector of slow variables x,vector functions X,vector of fast phases ψ,and vector of phase velocities Ω in explicit form

Furthermore,we use the algorithm of the modernized method of averaged characteristics[7,39]for the case of several fast phases.According to this algorithm we proceed to averaged variables

Equations for slow variables have the following form

We restrict ourselves by the third approximation for 1/ξ.The algorithm for finding out u(n)and A(n)is known and described,e.g.in[7,27,39].In particular,from these formulas it follows thatfor any n;for n ＞ 1;

and so on.As a result,we obtain the solution both for the oscillating velocity component and constant velocity component(13).

The continuity equation solution is obtained in the same way as in the case of the motion problem.To solve the electromagnetic field excitation problem,we substitute the expressions for the velocity(13)and the concentration of partial beams into Maxwell’s equation(8).We consider that these expressions obtained using the modernized method of averaged characteristics have the form of a series with the small parameter 1/ξ.We also expand it in a Fourier series in the fast phase harmonics.After a series of mathematical transformations,we get the system of self-consistent nonlinear differential equations for electric field strength harmonics complex amplitudes of the growing SCW in the cubic approximation

In(15)the harmonic number index m takes values from 1 to N,

is the SCW dispersion function,

The dependence of growth rate Γ on frequency ω with different average values of relativistic factors(curve 1 corresponds to γ01 = 4,curve 2—to γ02 = 5,curve 3—γ03 = 6)is shown in figure 5.The other parameters are the same as in the case of figure 2,curve 1.It can be seen that with the increase of γ0the increase of the multiharmonic SCW 1012s-1).Thus,as follows from figure 5,growth rates decrease,which should lead to the increase of the SCW saturation length.

The development of equations similar to(15)and(17)are described　in detail in　[7,27,28].FunctionsVq=contain cubic non-linear components.

Equation systems,consisting of(15)and(17),allow us to investigate multiharmonic processes in helical two-stream REB within the framework of cubic non-linear approximation.

One should note that equation systems,consisting of(15)and(17),describe both exponential growth of the SCW harmonics,because of the two-stream instability,and plural three-wave parametric resonance interactions between growing SCW harmonics.If the condition ωm ＜ ωcris fulfilled,then the dispersion equationDm　　(ω　　m　,k　m　)=0has complex roots

where iΓmis the imaginary part of the complex wavenumber.The quantity Γ is described as the growth rate,since this quantity de fines the exponential harmonic growth.Wave phases　pm　　=ω　m　t　-km 　zare defined only by the real part of the complex wavenumber.Therefore,the termDm　　(ω　　m　,km　　)⋅Emin(15)is also characterized by the real part of the complex wavenumber　Re　(k　compl,m　　)　=km and consequently is not equal to zero.If we retain only terms linear by Emin equation(15),the amplitude of growing the SCW can be written in the form Em ∝　exp　(Γ　m　z),then from(15)we get

and it is easy to find the growth rate

It is considered here,that as shown by numerical estimations.Thus,termsand DmEmin equation system(15)account for the exponential wave growth.

4.Analysis

Using equation system(15)–(17)we carry out the analysis of the spectrum form of the multiharmonic SCW depending on the two-stream REB parameters in the cubic non-linear approximation.We consider the case where only one SCW harmonic is excited upon the system input(z = 0).We take the first harmonic frequency ω1,which is much lower than the two-stream instability critical frequency ωcr.Then,as stated above,higher harmonics excitation and amplification occur due to the parametric resonant interactions in two-stream electron beam.These harmonics are also amplified due to the two-stream instability effect.Since the growth increments defined by two-stream instability are much higher than those defined by three-wave parametric resonance,the resulting amplification of the harmonics will be determined by the growth rates of the two-stream instability Γm.As a result,one should expect that the harmonics spectrum,or the dependence of the harmonic amplitude on the frequency at some value of longitudinal coordinate z,will be determined by the dependence of the growth rate on frequency Γ = Γ(ω).It is simple to achieve approximate analytical solutions of(16)[7,31,36],or to solve equation(16)numerically.Therefore,analyzing the growth increment dependence on frequency Γ = Γ(ω),we can estimate the frequency spectrum width ωcr - ω1of the multiharmonic SCW at different parameters of the two-stream REB.Furthermore,utilizing cubic non-linear equations(15)–(17)we determine the real frequency spectrum width of the multiharmonic SCW and saturation levels for different parameters of the two-stream REB.

Figure 2.Dependence of the growth rate Γ on frequency ω for the two-stream REB with different input angles α.The beam has the following parameters:The first harmonic frequency is Curve 1 corresponds to the case,curve 2—to the casecurve 3—to the case

Using the approach described above,we carry out the analysis of the　multiharmonic　SCW　spectrum　width dependence on the two-stream REB parameters.Let us find out the conditions of the maximal multiharmonic SCW spectrum width occurrence.We investigate the dependence of the multiharmonic SCW spectrum width on the helical twostream REB input angle α,relativistic factors average value γ0and difference Δγ,and plasma frequency of partial beams ωp.

Figure 2 represents the dependences of the growth rate Γ on frequency ω for the two-stream REB with different input angles α achieved as a result of numerical solution of dispersion equation(16).The beam parameters are the following: ωp = 6 × 1010s-1,average value of the partial beam relativistic factor γ0 = 4,difference between partial relativistic factors Δγ = 0.4,and the first harmonic frequency ω1 = 2.6 × 1011s-1.Curve 1 corresponds to the case α1 = 0°,curve 2—to the case α2 = 10°,curve 3—to the case α3 = 20°.It follows from figure 2 that for helical two-stream beams the critical frequency of the two-stream instability increasesω　cr1　＜　ωc　r2　＜　ωcr3with an increase of the beam input angle α.Optimal frequency ωopt,which corresponds to the maximal growth increment also increasesωo　pt1　＜　ωo　pt2＜ωo　pt3.The critical frequencyωc　r1　=　3.2　×　101　2　s-1corresponds to the input angle α1 = 0°,ωc　r2　=　3.8　×　101　2　s-1—to the angle α2 = 10°,and ωc　r3　=　5.2　×　101　2　s-1—to the angle α3 = 20°.　Therefore,　the　critical　frequency　increasestimes with an increase of the input angle from α1 = 0°to α3 = 20°.To estimate the multiharmonic SCW spectrum width,we take the difference between the critical frequency ωcrand first harmonic frequency ω1(ωcr - ω1).It follows from figure 2 that the multiharmonic SCW spectrum width increases with an increase of the input angle α.This means it is preferable to use helical two-stream REBs in multiharmonic FELs,the main purpose of which is to form a powerful multiharmonic signal with a broad frequency spectrum[1,21–24].It also follows from this figure that growth rates increase with an increase of the input angle.This means that the amplification rates of the SCW are higher in helical beams,which should lead to a decrease of the saturation lengths.

Figures 3(a)and(b)show the spectra of the multiharmonic SCW for the input angles α1 = 0°and α3 = 20°for the two-stream REB,with parameters that correspond to the case of figure 2 and are achieved from the cubic non-linear equation system(15),(17).We have considered 30 SCW harmonics in the simulation(N = 30).The first harmonic amplitude on the system input(z = 0)is 10 V cm-1,while other harmonics are zero.It is worth noting that electron beam parameters(beam input angle,relativistic factor,difference between relativistic factors,average value of the plasma frequency)have a Significant influence on the SCW saturation length(e.g. figure 4).The formation of the SCW with a broad frequency spectrum occurs in the saturation area.Therefore,SCW spectra in figure 3 and in subsequent figures are shown for the different coordinates.

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From comparison of figures 3(a)and(b),it follows that the multiharmonic SCW spectrum width in the case of helical two-stream REB is greater than in the case of the straight electron beam.We should note that the frequency at which the harmonic amplitude is minimal,ωminexceeds the two stream instability critical frequency ωcrin both cases α1 = 0°and α3 = 20°.As follows from figures 3(a)and(b),the real spectrum width should be defined by difference ωmin - ω1.One can see that the spectrum width increases～1.5 times with an increase of the beam input angle from α1 = 0°to α3 = 20°.

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As expected,maximal values of the SCW harmonic amplitudes in both cases are characterized by frequencies ωopt,which correspond to the maximal growth rates(see figure 2).This indicates that the two-stream instability effect prevails over the three-wave parametric resonant interactions during the formation of the multiharmonic SCW.

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It follows from figure 3(a)that there are non-zero harmonics with frequencies exceeding ωminpresent in the multiharmonic SCW spectrum.These harmonics are improper in this frequency domain and occur due to the parametric resonance(interference)of longitudinal waves.Interference components are also present in the spectrum of helical two stream REB(not shown in figure 3(b)).

Figure 4 shows electric field strength amplitudes of 30 SCW harmonics as functions of the longitudinal coordinate z for different beam input angles α = 0°(curves 1)and α = 20°(curves 2).Simulation parameters are the same as in the case of figure 3.In both cases,the SCW is monochromatic on the input with the frequencyω1=　2.6　×　101　1　s-1.It follows from figure 4 that amplification rates of the harmonics of electric field strength for the helical electron beam are higher(curves 2)than for the straight electron beam(curves 1).Consequently,SCW saturation in helical REBs(curves 2)occurs earlier than in the straight REBs(curves 1).It also follows from this figure that saturation levels in the case of the helical electron beam(curves 2)are～2.5 times lower than in the case of the straight beam(curves 1).The same conclusion follows from figures 3(a)and(b).

Figure 3.Dependence of the multiharmonic SCW harmonic amplitudes Emon frequency ω.Figure 3(а)shows the SCW spectrum in the straight two-stream REB(α1 = 0°)on z = 162 cm,while figure 3(b)shows the SCW spectrum in the helical two-stream REB(α3 = 20°)on z = 110 cm.Calculations are performed with the same parameters as in the case of figure 2.First harmonic amplitude on the system input(z = 0)is 10 V cm-1,while other harmonics are zero.First harmonic frequencyω1 =　2.6　×　101　1　s-1.

Figure 4.Electric field strength amplitudes of 30 SCW harmonics as functions of the longitudinal coordinate z for different beam input angles:α = 0°(curves 1)and α = 20°(curves 2).Simulation parameters are the same as in the case of figure 3.

Thus,two-stream superheterodyne FELs with helical electron beams can have smaller geometric dimensions due to higher amplification rates in comparison with two-stream superheterodyne FELs with straight beams.

are the functions considering the cubic non-linear terms and depending on the electric field strengths of the interacting waves.System(15)coefficients also depend on the constant components of velocities υqand concentrations nqof partial beams,which change during the non-linear interaction of the SCW harmonics.Therefore,we add the equations for constant components to system(15):

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Figure 5.Dependence of growth rate Γ on frequency ω with different average values of relativistic factors γ0.The beam has the following parameters:ωp=　6　×　101　0　s-1 ,Δγ = 0.4,input angle α = 0°.First harmonic frequency isω1 =2.6×1011s-1.Curve 1 corresponds to γ01 = 4,curve 2—to γ02 = 5,curve 3—γ03 = 6.

Figure 6 represents the spectrum of the multiharmonic SCW for the two-stream REB with γ02 = 5 at z = 268cm,with beam parameters the same as in the case of curve 2 in figure 5.The multiharmonic SCW spectrum for the average frequency spectrum width ωcr - ω1occurs.The biggest spectrum width is reached at the maximum value of the average relativistic factor,in this case γ03 = 6( figure 5,curve 3,ωc　r5　- ω 1=5.8 ×relativistic factor γ01 = 4 at z = 162cm,with the same beam parameters as in the case of curve 1 in figure 5,is shown in figure 3(a).Both the above mentioned figures have the first SCW harmonic amplitude of 10Vcm-1,while other harmonics are zero.The first harmonic frequency isω1=　2.6　×　101　1　s-1.

Figure 6.Dependence of the multiharmonic SCW harmonic amplitudes Emon frequency ω for the two-stream REB with γ02 = 5 at z = 268 cm,with the beam parameters that correspond to the case of curve 2 in figure 5(ωp=　6　×　101　0　s-1, Δγ = 0.4,α = 0°,ω1=2.6×1011s-1).

Comparing figures 3(a)and 6,we can conclude that with the increase of the average relativistic factor of the two-stream REB from γ01 = 4 to γ02 = 5 the multiharmonic SCW spectrum width ωmin - ω1increases 30%,as expected.Thus,the maximal level of the signal amplitude decreases 40%.As expected,coordinate z at which the multiharmonic spectrum forms for the system with γ02 = 5,increases to z = 268 cm compared with z = 162 cm for the system with γ01 = 4.Therefore,the increase of the relativistic factor of the twostream REB leads to the increase of the multiharmonic SCW frequency spectrum width.The formation of such a spectrum occurs at greater system lengths.

Figure 7 represents the dependence of the growth rate Γ on frequency ω at different values of the difference between the partial relativistic factors(curve 1 corresponds to Δγ1 = 0.4,curve 2—to Δγ2 = 0.3,curve 3 to Δγ3 = 0.2).It can be clearly seen that with decreasing of Δγ the increasing of the frequency spectrum width occurs.Thus,the maximal value of the growth rate remains practically the same,i.e.the formation of multiharmonic spectra should take place at comparable lengths.

The multiharmonic SCW spectrum for the two-stream REB with Δγ3 = 0.2 at z = 212 cm is shown in figure 8.The beam parameters for this spectrum correspond to the case of curve 3 in figure 7.The multiharmonic SCW spectrum with relativistic factor difference Δγ1 = 0.4 at z = 162 cm is shown in figure 3(a)(the beam parameters are the same as for curve 1 in figure 7).As expected,spectrum width ω　min,7　-ω1 at Δγ3 = 0.2( figure 8)is 1.6 times greater compared with the spectrum width ω　min,1　-ω1at Δγ4 = 0.4.Thus,the decrease of relativistic factor difference leads to a Significant increase of the multiharmonic SCW spectrum width.It also follows from the comparison of figures 8 and 3(a)that the saturation level decreases nearly three times with the relativistic factor difference decreasing to Δγ3 = 0.2.It should also be noted that the saturation length of the SCW increases 1.3 times for the beam with Δγ3 = 0.2.

Figure 7.Dependence of the growth rate Γ on frequency ω at different values of the difference between partial relativistic factorsThe beam has the following parameters:input angle α = 0°.First harmonic frequency isCurve 1 corresponds to the case curve curve 3 to

Figure 8.Dependence of the multiharmonic SCW harmonic amplitudes on frequency ω for the two-stream REB with,with the beam parameters that correspond to the case of curve 3 in figure 7

Figure 9 illustrates the growth rate Γ dependence on frequency ω at different values of partial plasma frequencies(curve 1 corresponds to the caseω　p1　=　6　×　101　0　s-　1,curve 2—to the caseω　p2　=　8　×　101　0　s-　1　,curve 3—to the case ωp3　=　10　×　101　0　s-1).The other parameters are the same as in the case of figure 2,curve 1.One can see that the multiharmonic SCW spectrum widthωc　r　-ω1increases with an increase of the ωp.The maximum value of the spectrum width is reached at the maximum value of the partial plasma frequency,in this case atω　p3　=　10　×　101　0　s-1( figure 9,curve 3).Moreover,as follows from figure 9,growth rates increase,which should lead to the reduction of the saturation lengths of the SCWs.

Figure 9.Dependence of the growth rate Γ on frequency ω at different values of partial plasma frequencies .The beam has the following parameters:,input angle α = 0°.First harmonic frequency isCurve 1 corresponds to the casecurve 2—to the casecurve 3—to the case.

Figure 10.Dependence of the multiharmonic SCW harmonic amplitudeson frequency ω for the two-stream REB withat z = 149 cm,with beam parameters that correspond to the case of curve 3 in figure 9

The multiharmonic SCW spectrum for two-stream REB withωp3　=　10　×　101　0　s-1atz = 149 cm　isshown　in figure 10.The beam parameters for this spectrum coincide with curve 3 in figure 9.The multiharmonic SCW spectrum with partial plasma frequencies ωp1　=　6　×　101　0　s-1　at z = 162 cm is shown in figure 3(a)(the beam parameters are the same as in the case of curve 1 in figure 9).As expected,spectrum widthω　min,9　-ω1atω　p3　=　10　⋅1　01　0　s-1( figure 10)is 1.5 times higher compared with spectrum widthω　min,1　-ω1 atωp1　=　6　⋅1　01　0　s-1.Therefore,partial plasma frequency increase of the two-stream REB leads to a Significant increase of the multiharmonic SCW spectrum width.It also follows from comparison of figures 10 and 3(a)that the saturation level increases 1.3 times with an increase of the partial plasma frequency up toωp3　=　10　×　101　0　s-1.Also,the length at which SCW saturation occurs decreases 1.1 times for the beam withωp3　=　10　×　101　0　s-1.

新媒体时代，内容为王。企业借助新媒体平台持续性地发布信息与大众进行日常对话，帮助企业提高大众对于品牌的认知，并利用精彩的内容帮助提升大众的好感度。及时利用新媒体平台主动发布企业信息澄清事实；还可以借助文字、图片、视频及其组合的形式，输送用户易于接受，甚至乐于传播的宣传信息来吸引关注；通过搜索引擎平台，将以企业为核心的相关关键词开展定制新闻，把握第一时间的新闻信息，为应对负面信息埋下应对基础。同时，通过对媒体信息的监控，有助于企业及时了解行业自身的业务信息、竞争对手动态、政策趋势、用户反馈等，以用于对当前公关活动成果的评估及后续行动的参考。

将探测器和Ellipse2-A固定在一起,进行姿态解算精度的分析和比较。通过串口将各自的姿态解算结果发送到上位机,使用MATLAB搭建姿态解算评估系统上位机,不仅可以保证探测器和Ellipse2-A同时采样,还可以对姿态解算结果和精度进行直观的实时显示和即时评价。

Therefore,in order to achieve the multiharmonic SCW with a broad frequency spectrum one should use dense high energy partial electron beams characterized by close relativistic factors.

由表2可以看出，该数据为平衡面板数据，截面数为31，跨期为11，属于短面板。被解释变量中，人均教育支出均等化指数均值为0.972，人均医疗卫生和人均社会保障与就业均等化指数均值都为0.999，接近于1，说明教育、医疗卫生和社会保障与就业基本公共服务均等化供给程度相对较高。

5.Conclusions

Therefore,we　developed　the　cubic　non-lineartheory describing the dynamics of the multiharmonic SCW,with harmonics frequencies smaller than the two-stream instability critical frequency,with different REB parameters.The selfconsistent differential equation system for the multiharmonic SCW harmonic amplitudes was elaborated in the cubic nonlinear approximation.This system considers both the twostream instability effect and the plural three-wave parametric resonances between the wave harmonics.We studied the mode of multiharmonic SCW formation,in which the first harmonics are much lower than the two-stream instability critical frequency.The different REB parameters such as the input angle with respect to the focusing magnetic field,average relativistic factor value,difference of partial relativistic factors,and plasma frequency of partial beams were analyzed regarding their influence on the frequency spectrum width and the multiharmonic SCW saturation levels.

We demonstrated that the multiharmonic SCW frequency spectrum width increases with an increase of the plasma frequency average value,the beam input angle α,the relativistic factor average value and with the decrease of the relativistic factor difference.The number of high-frequency harmonics increases in such an SCW spectrum.

We found out that saturation levels of the formed multiharmonic SCW decrease with an increase of the beam input angle α,the relativistic factor average value and with a decrease of the relativistic factor difference.In addition,amplitudes of the multiharmonic SCW increase with an increase of the plasma frequency.

We showed that growth rates increase with an increase of both the beam input angle α and plasma frequency.It is made clear that in this case the multiharmonic SCW saturation lengths decrease.Thus,utilizing such beams in two-stream superheterodyne FELs is expected to lead to a decrease of their longitudinal dimensions.

Therefore,in this article we have suggested ways of increasing the multiharmonic SCW frequency spectrum width in order to use these waves in multiharmonic two-stream superheterodyne FELs.The most effective with respect to increasing the multiharmonic SCW amplitude and frequency spectrum is the use of two-stream REBs with high electron density(plasma frequency).Despite the decrease of the multiharmonic SCW saturation levels in helical beams,they are of interest for such FELs,since systems with smaller longitudinal dimensions can be built on the basis of such beams.There is also interest in the increase of the SCW spectrum due to the change of relativistic factor and the partial relativistic factor difference,since the formation of the multiharmonic electromagnetic signal in two-stream superheterodyne FELs occurs not only due to the two-stream instability.The parametric resonance between the multiharmonic SCW,the pump and the electromagnetic signal wave plays a Significant role here.Therefore,the change of the relativistic factor and partial relativistic factor differences could enhance the effectiveness of the multiharmonic electromagnetic signal formation in twostream superheterodyne FELs.

This work was supported by the Ministry of Education and Science of Ukraine under Grant No.0117U002253.

1.加强罪犯劳动教育。要教育罪犯树立有劳动能力必须参加劳动的观念，为罪犯提供劳动岗位；要强化劳动组织管理，提高罪犯技术水平，积极开展技术革新，不断提高劳动效率，使罪犯了解市场经济对劳动者的技术水平和团结协作精神的要求；要依法保障罪犯的合法权益，为罪犯提供必需的劳动保护，依法科学合理安排劳动工时，严禁超时、超体力劳动。要充分调动罪犯参加劳动的积极性，使罪犯通过劳动掌握劳动技能，养成职业道德。

ORCID iDs

Iurii VOLK https://orcid.org/0000-0002-0262-762X

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