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В1СНИК ХНТУ №3 (66), ТОМ 2, 2 018 р.
УДК 53.01, 51.73
Ф УНДАМЕНТАЛЬШ НАУКИ
V.M. KUKLIN, D.N. LITVINOV, A.E. SPOROV
V.N. Karazin Kharkiv National University
THE SUPERRADIANCE OF MOVING AND STATIONARY OSCILLATORS
The mathematical model of superradiance of the system of nonlinear oscillators that are restricted in space is considered. The features of the synchronization process of such oscillators are discussed. A comparison of the superradiance efficiency of a system of interacting oscillators and a dissipative generation regime in a similar system is made, where the oscillators interacts only with the field of the excited wave. Also it was obtained the equations that describe the excitation of a TE wave by a beam of electrons rotating in an external magnetic field in a waveguide in different regimes, including the superradiance regime, where the beam electrons interact with each other due to their spontaneous radiation. The basic features of the description of the gyrotron gain regime are also discussed.
Keywords: interacting beam electrons, dissipative generation regime, superradiance regime, gyrotron gain regime, TE wave.
в.м. КУКЛ1Н, д.м. литвинов, o.e. споров
Харювський нацюнальнш ушверситет ÍMem В.Н. Каразша
НАДВИПРОМ1НЮВАННЯ РУХОМИХ ТА НЕРУХОМИХ ОСЦИЛЯТОР1В
У cmammi представлено математичну модель випромтювання системи нелтшних осциляторiв, обмежених у npo^opi. Розглянуто особливостi процесу синхронгзаци таких осциляторiв. Зроблено порiвняння ефективностi системи взаемодтчих осциляторiв в режимi надвипромтювання та в режимi дисипативно'1' генерацИ в аналогiчнiй системi, де осцилятори взаемодтть лише з полем збудженоi xerní. Також були оmриманi рiвняння, що описують збудження ТЕ xвилi в р1зних режимах пучком електротв, як обертаються у зовншньому магнтному полi хвилеводу, включаючи режим надвипромтювання, в якому електрони пучка взаемодтть один з одним завдяки ix спонтанному випромiнюванню. Також обговорено основнi особливостi опису режиму пiдсилення гiротрона.
Ключовi слова: взаемодiючi електрони пучка, режим дисипативной генерацИ, режим надвипромтювання, режим пiдсилення гiротрона, TE-хвиля.
В.М. КУКЛИН, Д.Н. ЛИТВИНОВ, А.Е. СПОРОВ
Харьковский национальный университет имени В.Н. Каразина
СВЕРХИЗЛУЧЕНИЕ ДВИЖУЩИХСЯ И НЕПОДВИЖНЫХ ОСЦИЛЛЯТОРОВ
В статье представлена математическая модель излучения системы нелинейных, ограниченных в пространстве осцилляторов. Рассмотрены особенности процесса синхронизации таких осцилляторов. Сделано сравнение эффективности системы взаимодействующих осцилляторов в режиме сверхизлучения и в режиме диссипативной генерации в аналогичной системе, где осцилляторы взаимодействуют только с полем возбужденной волны. Также были получены уравнения, описывающие возбуждение ТЕ волны в различных режимах пучком электронов, вращающимся во внешнем магнитном поле волновода, включая режим сверхизлучения, в котором электроны пучка взаимодействуют друг с другом за счет их спонтанного излучения. Также обсуждены основные особенности описания режима усиления гиротрона.
Ключевые слова: взаимодействующие электроны пучка, режим диссипативной генерации, режим сверхизлучения, режим усиления гиротрона, TE-волна.
Formulation of the problem
Firstly Dicke has found in his work [1] that taking into account the interaction of two-level atoms in a small volume leads to the acceleration of the relaxation processes that remove their excitation and synchronization of radiation. This effect was called superradiance. In the classical case, one can also take into account the interaction of radiating particles, because each of them interacts with others through its field. In the absence of an external field, such an interaction can lead to a certain phase (and, in some cases, spatial) synchronization of oscillators and the appearance of a coherent component in the integral radiation of the system [2].
Analysis of recent research and publications
In the work [3], the nature of the radiation of Langmuir oscillations both by extensive and by comparable with the wavelength of non-relativistic electron beams of has been considered. The latter case corresponds to the superradiation of a cluster beam and qualitatively corresponds to the dissipative generation regime. In this case, the process of the synchronizing of oscillators, creating the conditions for the formation of coherent radiation, was
В1СНИКХНТУ№3(66), ТОМ2, 2018 р._ФУНДАМЕНТАЛЬШНАУКИ
determined by their spatial grouping. However, for such particles whose centers of oscillations are stationary the spatial grouping is impossible and the phase synchronization is hampered [4 - 7] and this type of nonlinear oscillators can be synchronized with the help of an external small amplitude field [5 - 7].
The aim of research
The main aim of this article is to find the relationship between descriptions in different physical realizations of oscillators whose centers are fixed. Also it is important to make a comparison between the superradiance efficiency of a system of interacting oscillators and the dissipative generation regime in a similar system where the oscillators interact only with the field of the excited wave. As well this work is aimed to the problem of radiation of the group of interaction with each other electrons under the conditions of the cyclotron resonance with a TE wave.
Main part
Wake emission of the moving short beam of electrons in plasma
The propagation of a short electron beam, with the length b and the unperturbed velocity in plasma v0 in
the simplest one-dimensional case can be described by the following system of equations [2, 8]:
N
2n ■ /dT = -(Nв)-1£cos [2n g¡ (Zj)] U(Z, -Zj), (1)
i=1
where 2nZ = kz -mt, т = t ■ у, mpe ь = 4ne2n0e ь / me, e,me - are the electron charge and mass, respectively, n0e ь - are the plasma and beam density; the oscillation effective decrement in the beam SD can be determined as the ratio of the energy flux of the oscillations leaving the beam volume to the total energy of the oscillations in its volume SD = Vo/a, e = SD / T\s=o = V0/a ■ m pe )m / m^ )-2/3,Y = («0b / n0e)' ®le, M = npb'b - is the total number of particles in the beam, g = (1 + (v - v0)/ v0)-1 = (1 +Vmpe / y)-1, that can be easily determined from the
equation 2ndZ/dT= k (v - vj) / Y = V. Here и (x) = 1; x > 0 and U (x) = 0; x < 0. The parameter 9 = Sd / Y corresponds to the ratio of the oscillation damping decrement in the absence of a nonequilibrium element (here this is the beam) 5d to y - the maximum increment of nondissipative instability (in the absence of losses).
The right-hand side of (1) describes the total field of particles of a sufficiently short beam, for large в values, that is, it determines the sum of the spontaneous Langmuir fields with the frequency mpe that are radiated
by the individual particles of the beam.
For large values в >> 1, the interaction of particles with each other can be significant (the particles lies ahead in the direction of motion act on the following ones, but not vice versa). When the beam particles are grouped and at the same time the phase synchronization takes place, the formation of so-called superradiance is possible.
If the beam is rather short, then the field energy in the time v0 / b << y_1 (that is, в >> 1) is carried out from the volume of the bunch. In this case, the field growth in the beam volume is occur due to the independent grouping of particles and the increase of the coherence of their radiation, which forms the superradiance field. It was noted in [2, 8] that the increment of the superradiation process of a beam bounded in the direction of motion is comparable with the increment of the dissipative beam instability yd ~ mpb (mpe / Sd )12 = mpb (kb)1/2, and the energy loss in
which is determined by the energy transfer from the beam volume. That is, the spatial modulation of particles, that is similar in both cases, lead to the phase synchronization of the particles and, accordingly, to the magnification of the radiation coherence. The limitation of the amplitude was occurred due to the capturing of the particles in the field of the wave that corresponds to the equality of the frequency of the oscillations of trapped particles in the potential well of the wave to the increment of the process Q = >/ekE / m « yd . The maximum value of the field amplitude for the beams, whose length is in the several times larger than the length of the Langmuir wave, is equal to Emax « 2neM and depends only on the number of particles.
It is easy to obtain that the maximum radiation intensity in the superradiance regime for beams with the length about several times larger than the length of the Langmuir wave reaches the values
2 ne2v 2 2
^sup = v ■ Emax / 4n = —-—M x M , that is, proportional to the square of the total number of the particles in
the bunch M = щ ■b . Let us note that the intensity of the spontaneous emission of the same bunch of particles is
2 2
about Psp0n = v ■ Esp0n / 4n = ne vM x M and is proportional to the total number of particles.
The system of oscillators with the fixed centers Let us consider the processes of electromagnetic waves generation by a system of oscillators with the fixed centers. Let consider the case when the wave frequency and the oscillators frequency coincides and equal to m .
В1СНИКХНТУ№3(66), ТОМ2, 2018р._ФУНДАМЕНТАЛЬШНАУКИ
The wave vector of the oscillations is k = (0,0,k), the wave components are E = (£,0,0), B = (0,E,0), where E =| E | • exp{-i< + ikz + iq>} . The oscillators are placed along the OZ axis with quantity of N per wave length 2n / k. The oscillator's mass is m, the charge is equal to -e and oscillator's frequency coincides with wave frequency < . The oscillator's initial amplitude is equal to % . Let assume that the oscillator moves only along the axis OX . In this case, the influence of the magnetic field of the wave on the oscillator dynamics can be neglected [5].
For the extended systems, or for the case of small group velocity of the excited oscillations, it becomes possible to store the field energy in the active zone, even at a finite level of radiation losses. Further, we will assume that the system is rather extended, and the group velocity of the wave is small, for example, because of the effective
2
permittivity (cef = Vg = ^c / <0S0 ) or near the boundary frequency of the waveguide, as, for example, realized in gyrotrons. Also, the reflection from the boundaries of the system of oscillators is neglected. In this case the effective decrement of absorption is SD = 2c f / b and 6 = 8/ y = 2cejf / b • Yi. In general, it is necessary to
take into account electromagnetic waves of induced radiation, which propagate in both directions. As for the spontaneous radiation of each particle, and of the oscillator system, it is always oriented in both directions. The equations of the oscillators motion can take the form:
d 1 N (2)
—Aj = -—X As ■ [Cos{<^ + 2n(Zj -Zs) + U+ (Zj - Zs)- Cos^j - 2n(Z j -Zs) + ^S]U+(ZS - Zj)] - E0 • Cos{2nZj }, ^
1 s=1
d 1 N (3)
Aj [—Vj - A,] = -—X As -[Sin{^j + 2n(Zj - Zs) + Vs}• U + (Zj - Zs) - Sin{^. - 2n(Zj - Zs) + }• U+ (Zs - Zj)]- E0 • Sin{2nZj}. (3)
1 s=1
The right-hand sides of equations (2) and (3) contain the normalized values of the amplitude of the total field of the spontaneous radiation of its neighbors and the external synchronizing field acting on the oscillator.
Taking the relativism into account leads to the non-linearity of the oscillators A, =a • (A,2 -A,20),
2
a = 3<(k• oq) / 4. It was obtained that for the waves of different polarization (when the particles move only along the direction of the electric wave field vector), the same equations can be used. Let us define the conditional operator U+ (z -zs) = 1 when z > zs , and U+ (z - zs) = 0 when z < zs , the density of the particles in a unit section per unit length и0 = M / b , where M is the total number of particles on the length of the bunch, and each model
particle contains M / N real particles, ne2M / 2mc = y2 / 8 = yD is the square of the increment (inverse characteristic time of development) of the process in both cases (i.e., superradiance and dissipative instability regimes for the same 6 = 8/ y>> 1 values), and 8 = 2cejf / b is the effective decrement of radiation losses in the
system. It was earlier noted in [6] that for some 6 values the characteristic times of the development of the superradiance process and the dissipative instability process of the system of oscillators coincide. Moreover the maximum possible radiation intensity levels of a spatially confined particle beam were coincided.
It is also interest to determine the efficiency of the superradiance regime of a system of oscillators and to compare it with the regime of dissipative instability considered earlier in [7].
The results of the numerical investigation are presented on Figs. 1-5. For the case when the number of oscillators N = 10 000, the average integral field | E | of a system of oscillators with a random distribution of the initial phase is in 100 times smaller than the maximum possible value of the field in the case when all the oscillators are synchronized in phase. That is, the spontaneous and absolutely coherent induced radiation satisfies the relation
—2 I Z7 12 - 4
10 -1. For the squares of the amplitudes | E | , this relation takes the form 10 -1.
In the absence of a external synchronizing field with amplitude E0 = 0, one can observe a significant scatter of the maximum values of the radiation field at the ends of the system whose size is equal to the wavelength of the radiation AZ = 1. For rather great nonlinearity of the oscillators a = 1, the squares of the maximum amplitudes | E |2 become to be the largest and change for random distributions of the initial phases in the range from 0.04 to 0.08. The increasing of the system size reduces the spreading of the maximum values for any nonlinearity values, but it also reduces the maximum possible radiation amplitude at the ends of the system.
The influence of an external field that only in five times greater than the average spontaneous level
E0 = 0.05 leads to a sharp decrease of the spreading of the maximum values of the field at the boundaries of the
2
system. So, for the size of the system AZ = 1 for different values of nonlinearity, the values | E | are shown in Fig. 1.
BICHHK XH TV №3 (66), TOM 2, 2018 p.
0 VHJA M EHTÂllbH I HÂYKH
Fig.1. The values of
a .
depending of the level on nonlinearity
0 5 10 15 20 25 30 35 40 45 Fig. 2. The dependence of the field to the right of the system, to the left of the system and the maximum within the system upon the time, AZ = 1, a = 1.
Thus, it can be concluded that the external field has a stabilizing effect on the generation pattern. The greatest value of the field in the opposite side of the system according to the direction of propagation of the external
2
wave | E |« 0.23 * 0.25 and | E p « 0.08, that is, the degree of coherence is of 8%.
In the paper [7] the behavior of the dissipative generation regime of the system (2) - (3) is analyzed. It is shown that for values 0 « 10 the maximum of the energy radiation from the system is reached (that is, the value
2 2 O | E | is maximal) and at the same time | E | reaches the same values as in the case of superradiation. That is,
the degree of coherence of the radiation of the oscillator system is comparable.
0 5 10 15 20 25 30 35 40 45 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
Fig. 3. The normalized total energy of the particles as a function of Fig. 4. The amplitude of the oscillators at different points along Z time, AZ = 1, a = 1. at the moment of reaching the maximum of the field, AZ = 1,
a = 1.
0 0,1 0.2 0.3 0,4 0,5 0,6 0,7 0,8 0,9 1 Fig. 5. The phase of the oscillators at different points along Z (to the right) at the moment of reaching the maximum of the field,
AZ = 1 ,a = 1.
В1СНИКХНТУ№3(66), ТОМ2) 2018р._ФУНДАМЕНТАЛЬШНАУКИ
Conclusions
In the case of radiation from a short beam - bunch of electrons moving in plasma the superradiance regime is realized. At that a wake is formed behind the bunch [2, 3]. The beam particles interact with each other due to the fields radiated by them. This circumstance made it possible to consider this process as superradiance [2]. One can consider the generation of radiation by beam particles (whose density and velocity are similar to the case discussed above) in a system of the same dimensions. While the radiation energy extracts from the ends of such short system, the so-called dissipative regime of beam instability is realized. Usually in this case the particles spontaneous radiation is neglected, thereby excluding the direct interaction of the particles with each other. Only the interaction of particles with the fields of the wave (or of the wave packet) is taken into account. Nevertheless, the increments of the processes are the same and the achievable amplitudes of the generated fields are of the same order. It is possible to estimate the degree of coherence of the oscillators in relation to the achievable field intensity in the system to the maximum possible intensity of ideally phased radiation sources. In this research, this value reaches, as a rule, 25%.
In this paper, we discuss an analogous problem of elucidating the radiation efficiency of a short system of nonlinear oscillators. Here, is also possible to realize the regime of the interaction of the oscillators with each other due to the intrinsic (spontaneous) fields of their radiation. This corresponds to the superradiance regime and the generation regime at the same level of radiation absorption due to the limited system. This regime is the dissipative regime that was discussed in [7]. In this case, spontaneous radiation is neglected and the interaction of the oscillators occurs only with the wave fields, generally, propagating in both directions. It is also shown here that the increments of processes at the same level of radiation absorption are the same, and the maximum possible amplitudes are of the same order. The degree of coherence of the oscillators that can be estimated from the ratio of the achievable field intensity in the system to the maximum possible intensity of perfectly-phased radiation sources, reaches 8% here. It is important to note that, for better repeatability of the results, an external synchronizing wave should be used (for example, in the same way as suggested in [9]), the intensity of which exceeds the integrated spectral noise of un-phased oscillators. In the case discussed above, the noise intensity was about 0.01% and the intensity of the external field is 0,25% with respect to the maximum possible intensity of perfectly-phased sources.
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