Statistical fluctuations in cooperative cyclotron radiation

Nuclear Inst. and Methods in Physics Research, A 879 (2018) 77–83

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Nuclear Inst. and Methods in Physics Research, A journal homepage: www.elsevier.com/locate/nima

Statistical fluctuations in cooperative cyclotron radiation S.V. Anishchenko *, V.G. Baryshevsky Research Institute for Nuclear Problems, Bobruiskaya str., 11, 220030, Minsk, Belarus

a r t i c l e

i n f o

Keywords: Autophasing time Cooperative cyclotron radiation Superradiance SASE Shot noise

a b s t r a c t Shot noise is the cause of statistical fluctuations in cooperative cyclotron radiation generated by an ensemble of electrons oscillating in magnetic field. Autophasing time – the time required for the cooperative cyclotron radiation power to peak – is the critical parameter characterizing the dynamics of electron-oscillators interacting via the radiation field. It is shown that premodulation of charged particles leads to a considerable narrowing of the autophasing time distribution function for which the analytic expression is obtained. When the number of particles 𝑁𝑒 exceeds a certain value that depends on the degree to which the particles have been premodulated, the relative root-meansquare deviation (RMSD) of the autophasing time 𝛿𝑇 changes from a logarithmic dependence on 𝑁𝑒 (𝛿𝑇 ∼ 1∕ ln 𝑁𝑒 ) √ to square-root (𝛿𝑇 ∼ 1∕ 𝑁𝑒 ). A slight energy spread (∼4%) results in a twofold drop of the maximum attainable power of cooperative cyclotron radiation. © 2017 Elsevier B.V. All rights reserved.

1. Introduction A considerable attention has been paid recently to the generation of short superradiance pulses [1,2], also termed as ‘‘self-amplified spontaneous emission’’ or ‘‘cooperative radiation’’ [3], using short electron beams propagating in complex electrodynamical structures (undulators, resonators, dielectric and corrugated waveguides, or photonic crystals) [1,2,4–6]. The possibility to produce short pulses of cooperative radiation was first substantiated in [7,8] for free electron lasers (FELs). However, the experimentally measured output characteristics of cooperative radiation (power, radiation spectra, and energy in a pulse) have seemed really stochastic. According to theoretical [9,10] and experimental [11– 15] studies, statistical properties of cooperative radiation depend on the shot noise which is inherent to charged-particle beams. It is noteworthy that frequency and time stability of the radiation pulse is crucial for many applications. For this reason, electron beams are premodulated at radiation frequency to reduce the influence of shot noise on cooperative radiation [16–20]. In a microwave range, short-pulse cyclotron resonance masers (CRMs) [21–23] and Cherenkov generators [24–26] are more commonly used. The generators of cooperative radiation operate in two regimes: traveling wave [24–26] and backward wave [27–31]. The distinguishing feature of the traveling-wave regime is that the group velocity of electromagnetic waves and the velocity of charged particles are codirectional. This regime was applied in the first experiments with

short-pulse CRMs [22,23] and Cherenkov generators [24]. Theoretical consideration of radiation gain in short-pulse traveling-wave generators revealed an important peculiarity: the stability of the cooperative radiation parameters can be improved noticeably by injecting into the electrodynamical structure of a beam with a sharp front whose duration is comparable with the wave period [25,26]. In this case, the Fourier transform of the beam current contains quite a significant spectral component at radiation frequency. As a result, the generation of electromagnetic oscillations starts with coherent spontaneous emission of the whole beam instead of incoherent spontaneous emission from individual particles. As a result, the degree of fluctuations in cooperative radiation is decreased. Though the early short-pulse CRMs and Cherenkov generators operated in the traveling-wave regime, the most impressive results were obtained in experiments with short-pulse backward-wave tubes generating cooperative radiation whose peak power was appreciably greater than the beam power [29–31]. For example, the peak radiated power of 1.2 GW was attained at 9.3 GHz for beam power of 0.87 GW [29]. The stability of output parameters of the cooperative-radiation pulse generated in the backward-wave regime, as in a short-pulse travelingwave generator, strongly depended on the beam front. Analysis of theoretical and experimental works on generation of cooperative radiation reveals that the main source of statistical spread of the output characteristics is the shot noise which is inherent to electron

* Corresponding author.

E-mail addresses: [email protected] (S.V. Anishchenko), [email protected] (V.G. Baryshevsky). https://doi.org/10.1016/j.nima.2017.10.011 Received 20 June 2017; Received in revised form 5 October 2017; Accepted 6 October 2017 0168-9002/© 2017 Elsevier B.V. All rights reserved.

S.V. Anishchenko, V.G. Baryshevsky

Nuclear Inst. and Methods in Physics Research, A 879 (2018) 77–83

then the radiation reaction force acting on each particle is given by the sum [41]

beams. The impact, which the shot noise produces, can be diminished by beam premodulation at radiation frequency [16–20]. Naturally, one could wonder what degree of beam premodulation is required for certain applications. This problem would be most prevalent for high-current generators with electron beams composed of ectons, individual portions of electrons that contain up to 1011 elementary charge carriers [32,33]. To estimate the shot noise accurately, one must take into account the complex structure of electron beams [34]. This is essential when solving the problem of coherent summation of electromagnetic oscillations from several short-pulse sources of radiation [5,6]. This paper considers the effect of shot noise on the generation of cooperative cyclotron radiation from a premodulated short beam of particles. We shall consider this issue by the example of an ensemble of nonisochronous electron-oscillators interacting with one another via the radiation field, used as one of the simplest models for the description of nonlinear generation of electromagnetic waves in CRMs [35–40]. The term ‘‘nonisochronous’’ is related to oscillators with amplitudedependent frequencies [35]. It has been shown in [37,38] that in the absence of external action, the instability evolves in the ensemble of nonisochronous electron-oscillators, accompanied at the initial stage by an exponential growth of the radiated power and autophasing of electron-oscillators. This exponential growth is then suppressed due to nonlinearity, and the pulse of cooperative cyclotron radiation is formed [37]. The influence of nonisochronism on the peak power of cooperative radiation and the autophasing time of electron-oscillators was studied in detail by Vainshtein and colleagues [38]. However, the authors of [38] assumed that the beam had been premodulated at the radiation frequency and left out the effects related to shot noise caused by spread in positions and velocities of electrons [40]. The velocity spread is usually associated with the thermal nature of electron emission from the cathode surface. The position spread is originated from fortuitousness of the time moments of electron emission. In our analysis of statistical properties of cooperative cyclotron radiation, the peak radiated power and autophasing time serve as random variables. The paper’s outline is as follows. First, we derive a system of equations describing the interaction of nonisochronous oscillators via the radiation field by the example of electron ensemble circulating in a uniform magnetic field. Further comes the consideration of statistical fluctuations in cooperative cyclotron radiation in the presence of shot noise from the ensemble of nonisochronous electron-oscillators with and without phase premodulation of charged particles. Particular attention is given to finding the autophasing time distribution function.

𝐹⃗𝑘 = 𝑒

𝑚2 𝑐 4 ≈ 6 ⋅ 1015 Gs. 𝑒3

𝐻≪

(4)

But in the case of a dense beam of coherently emitting particle with 𝑁𝑒 electrons, mass 𝑀 = 𝑁𝑒 𝑚, and charge 𝑄 = 𝑁𝑒 𝑒 we can write by analogy with (4) 𝐻 ≪ 𝐻𝑐𝑟 =

𝑀 2𝑐4 𝑚2 𝑐 4 = . 𝑄3 𝑁 𝑒 𝑒3

(5)

With the present-day acceleration facilities, dense beams with 𝑁𝑒 ∼ 1010 electrons are available; substitution of 𝑁𝑒 ∼ 1010 into (5) gives 𝐻 ≪ 60 kGs. If this condition is violated, the equation set (1) with the force (3) is inapplicable. Thus, if the radiation reaction force is much less than the Lorentz force and the beam size is less than the radiation wavelength, then the equations of motion describing the interaction between charged particles have the form: 2𝑒2 𝑁𝑒 ̈ 𝑒 ⃗ + 𝑣⃗𝑘 × 𝐻 𝑣⃗⟂ , 𝑐 3𝑐 3 ∑ 1 𝑣⃗̈ ⟂ = 𝑣⃗̈ . 𝑁𝑒 𝑘 ⟂𝑘

𝑝̇ ⟂𝑘 =

(6)

In the absence of energy losses through emission, the particles are in circular motion with cyclic frequencies [41] √ ( 𝑣2 + 𝑣2𝑧𝑘 ) 𝑣2 𝑒𝐻 𝑒𝐻 1 − ⟂𝑘 , (7) 𝛺𝑘 = 1− 𝑘 ≈ 𝑚𝑐 𝑚𝑐 𝑐2 2𝑐 2 depending on 𝑣⃗⟂𝑘 , which is responsible for nonisochronism of oscillations. Using the approximate relation 𝑣⃗̈ ⟂𝑘 ≈ −𝛺𝑘2 𝑣⃗⟂𝑘 ≈ −𝛺2 𝑣⃗⟂𝑘 ,

Let us consider the behavior of a weakly relativistic electron beam in ⃗ directed along the 𝑂𝑍 axis in the presence a uniform magnetic field 𝐻 of the radiative energy loss. Particle velocity components perpendicular and parallel to the magnetic-field vector are denoted by 𝑣⃗⟂𝑘 and 𝑣⃗𝑧𝑘 , respectively. Then the behavior of particles is described by the equations of motion in the form

(8)

where 𝛺=

𝑒𝐻 , 𝑚𝑐

(9)

we shall write vector equations (6) in a component-wise fashion 2 2 ) ( 2𝑒2 𝛺2 𝑁𝑒 1 𝑣⃗⟂𝑘 + 𝑣⃗𝑧𝑘 𝑣̇ 𝑥𝑘 = 𝛺 1 − 𝑣𝑦𝑘 − 𝑣𝑥 , 2 2 𝑐 3𝑚𝑐 3 2 2 ) ( 2𝑒2 𝛺2 𝑁𝑒 1 𝑣⃗𝑘 + 𝑣⃗𝑧𝑘 𝑣̇ 𝑦𝑘 = −𝛺 1 − 𝑣𝑥𝑘 − 𝑣𝑦 , 2 2 𝑐 3𝑚𝑐 3

𝑒 ⃗ + 𝐹⃗⟂𝑘 , 𝑝⃗̇ ⟂𝑘 = 𝑣⃗⟂𝑘 × 𝐻 𝑐 (1)

Here, 𝐹⃗𝑘 is the radiation reaction force acting on the 𝑘th particle from all particles and 𝑝⃗𝑘 is its momentum related to the velocity 𝑣⃗𝑘 as ( 𝑣2 + 𝑣2𝑧𝑘 ) 𝑚𝑣⃗𝑘 𝑝⃗𝑘 = √ ≈ 𝑚𝑣⃗𝑘 1 + ⟂𝑘 . 2𝑐 2 1 − 𝑣2𝑘 ∕𝑐 2

(3)

where 3𝑐2𝑒3 𝑣⃗̈ 𝑗 is the radiation field induced by the 𝑗th particle. Let us pay attention to an essential circumstance [41]: the expression for 𝐹⃗𝑘 is true if the radiative friction force is appreciably less than the ⃗ acting on each particle. Otherwise, unphysical Lorentz force 𝑐𝑒 𝑣⃗𝑘 × 𝐻 self-accelerated solutions may arise. The requirement that the radiative friction force should be much less than the Lorentz force imposes limitation on the magnetic field strength (in the opposite case the considered theory is invalid). For single particle [41]:

2. Cooperative cyclotron radiation

𝑝̇ 𝑧𝑘 = 𝐹𝑧𝑘 .

∑ 2𝑒 𝑣⃗̈ , 3 𝑗 𝑗 3𝑐

𝑣̇ 𝑧𝑘 = 0,

(10)

where (2)

1 ∑ 𝑣⃗ , 𝑁𝑒 𝑘 𝑥𝑘 1 ∑ 𝑣⃗𝑦 = 𝑣⃗ . 𝑁𝑒 𝑘 𝑦𝑘

𝑣⃗𝑥 =

If the beam size is less than the radiation wavelength 𝜆 = 2𝜋𝑚𝑐 2 ∕𝑒𝐻, and the Coulomb repulsion force and induction fields can be neglected, 78

(11)

S.V. Anishchenko, V.G. Baryshevsky

Nuclear Inst. and Methods in Physics Research, A 879 (2018) 77–83

3. Shot noise

Then we shall multiply the second equation (10) by −𝑖 and add it to the first one: 2 2 ) ( 1 𝑣⃗⟂𝑘 + 𝑣⃗𝑧𝑘 𝑣̇ 𝑥𝑘 − 𝑖𝑣̇ 𝑦𝑘 = 𝑖𝛺 1 − (𝑣𝑥𝑘 − 𝑖𝑣𝑦𝑘 ) 2 2 𝑐 2𝑒2 𝛺2 𝑁𝑒 − (𝑣𝑥 − 𝑖𝑣𝑦 ). (12) 3𝑚𝑐 3 Assuming that all 𝑣𝑧𝑘 = 𝑣𝑧 are equal, let us introduce the following ∑ 2 2 notation: 𝑏𝑘 = 𝑒−𝑖𝛺(1−𝑣𝑧 ∕2𝑐 )𝑡 (𝑣𝑥𝑘 − 𝑖𝑣𝑦𝑘 )∕𝑣⟂0 (𝑣⟂0 = 𝑘 𝑣⟂𝑘 (0)∕𝑁𝑒 is the average tangential speed of particles). Then (12) takes the form [37]

In the absence of energy spread, shot noise in the ensemble of electron-oscillators is due to a random distribution of the initial phases 𝜙𝑘 (0). Each set of 𝜙𝑘 (0) corresponds to different initial conditions of the equation set (21). Fluctuations in the initial conditions lead to fluctuations in the output characteristics of cooperative cyclotron radiation which we study by a numerical experiment. As follows from (21), the behavior of the ensemble of nonisochronous oscillators in the absence of the energy spread is determined by two fixed parameters 𝜃 and 𝑁𝑒 and a random set of initial phases 𝜙𝑘 (0). Because the distribution of initial phases 𝜙𝑘 (0) is random, the numerical experiment with each pair of values of 𝜃 and 𝑁𝑒 must have many runs. This procedure will give information about statistical characteristics of cooperative radiation, the most important of which are peak power 𝑃0 , autophasing time 𝑇0 , and their relative RMSD — 𝛿𝑃 and 𝛿𝑇 . In numerical analysis of statistical fluctuations in cooperative cyclotron radiation in the presence of shot noise, instead of the number 𝑁𝑒 of real electrons we took the number 𝑁 = 36 ≪ 𝑁𝑒 of simulated electrons with the charge 𝑒𝑁𝑒 ∕𝑁 and initial phases √ 12𝑁 2𝜋𝑘 + 𝑟 , 𝑘 = 1 … 𝑁, (24) 𝜙𝑘 (0) = 𝑁 𝑁𝑒 𝑘

𝑑𝑏𝑘 + 𝑖𝜃|𝑏𝑘 |2 𝑏𝑘 = −𝑏, 𝑑𝜏 1 ∑ 𝑏= 𝑏 , 𝑁𝑒 𝑘 𝑘 𝜃=

3𝑚𝑣2⟂0 𝑐 4𝑒2 𝛺𝑁𝑒

,

2𝑒2 𝛺2 𝑁

𝑒 𝑡. (13) 3𝑚𝑐 3 The equation set (13) fully describes the behavior of the ensemble of electron-oscillators moving in a uniform magnetic field in the presence of the radiative energy loss. The kinetic energy 𝐸𝑟𝑎𝑑 of transverse motion of the oscillators, the radiation power 𝑃𝑟𝑎𝑑 , and the time 𝑡 are related to the dimensionless quantities by 1 ∑ 𝐸𝑟𝑎𝑑 ∕𝐸𝑈 = |𝑏 |2 , (14) 𝑁𝑒 𝑘 𝑘

𝜏=

𝑃𝑟𝑎𝑑 ∕𝑃𝑈 = 2|𝑏|2 ,

where 𝑟𝑘 are random variables uniformly distributed over the interval [0; 1). It has been shown in [42] that this procedure, boosting the performance of the program, correctly simulates the shot noise in the absence of energy spread. We selected the following values of controlling parameters: 𝑁𝑒 = 6.75 ⋅ 104 , 1.08 ⋅ 106 , 𝜃 = 1–107. The numerical experiment with each 𝑁𝑒 –𝜃 pair was repeated one hundred times. Figs. 1 and 2 show the results of our computation from which we can draw some very important conclusions [40]. First, the relative RMSD of the dimensionless peak radiated power weakly dependent on the √ number of particles 𝑁𝑒 (Fig. 1) decreases as 𝛿𝑃 ≈ 4.3∕ 𝑁𝑒 . Second, the autophasing time decreasing as the nonisochronism parameter 𝜃 is increased depends logarithmically on the number of particles 𝑇0 ∼ ln 𝑁𝑒 . Third, 𝛿𝑇 reduces according to the approximate formula 𝛿𝑇 = 𝑞∕ ln 𝑁𝑒 , where 𝑞(𝜃) ≈ 1.1 slowly decreases as lg(𝜃) varies from 0 to 2. Extrapolating the obtained dependence of 𝛿𝑃 (𝑁𝑒 ) and 𝛿𝑇 (𝑁𝑒 ) to the region with large number of particles, 𝑁𝑒 = 109 –1012 , (typical number of electrons in short-pulse generators), we get the estimates 𝛿𝑇 = 0.04–0.05 and 𝛿𝑃 < 10−4 . From this we can deduce that shot noise leads to a 4–5% fluctuation of the autophasing time at insignificant fluctuations of the peak radiated power. Let us note here that phase premodulation of electron-oscillators: √ ( ) 2𝜋𝑘 12𝑁 2𝜋𝑘 𝜙𝑘 (0) = + 𝑟𝑘 + 𝛿𝜙 cos , 𝑘 = 1 … 𝑁, (25) 𝑁 𝑁𝑒 𝑁

(15)

and (16)

𝑡∕𝑇𝑈 = 𝜏, where 𝐸𝑈 =

𝑃𝑈 =

2 𝑁𝑒 𝑚𝑐 2 𝑣⟂0 , 2 𝑐2

2𝑒4 𝐻 2 𝑁𝑒2 𝑣2⟂0 3𝑚2 𝑐 2

𝑐2

(17)

,

(18)

and 𝑇𝑈 =

3𝑚3 𝑐 5 2𝜃𝑐 2 . = 4 2 2𝑒 𝐻 𝑁𝑒 𝛺𝑣2⟂0

(19)

Let 𝐻 = 6.4 kGs, 𝑣∕𝑐 = 0.38, and 𝑁𝑒 = 109 , then 𝐸𝑈 = 6 μJ, 𝑃𝑈 = 0.5 kW, 𝑇𝑈 = 13 ns, and 𝛺 = 18 GHz. In this case, the nonisochronism parameter is 𝜃=

2 3𝑚2 𝑐 4 𝑣⟂0 = 100. 3 4𝑒 𝐻𝑁𝑒 𝑐 2

(20)

having practically no effect on 𝑃0 , 𝑇0 , and 𝛿𝑃 , leads to a noticeable decrease in the fluctuation of autophasing √ time: 𝛿𝑇 no longer decreases logarithmically (𝛿𝑇 ∼ 1∕ ln 𝑁𝑒 ), but as 1∕ 𝑁𝑒 (Fig. 3). In the expression (25), 𝛿𝜙 ≪ 1 is the premodulation parameter. One important thing should be noted. To calculate 𝑇0 , we determine the time 𝑇𝑖 when the maximum power is reached on the time mesh. Then, using three points (𝑇𝑖−1 , 𝑇𝑖 , and 𝑇𝑖+1 ), we construct the secondorder polynomial 𝑃 (𝑇 ) and find the point 𝑇0 where 𝑃 (𝑇 ) takes maximum value in the interval (𝑇𝑖−1 ;𝑇𝑖+1 ).

Using amplitude transformation 𝑎𝑘 = 𝑏𝑘 𝑒𝑖𝜃𝑡 , let us rewrite the equation set describing the behavior of the ensemble of weakly nonlinear electron-oscillators (13) in the following form [38]: 𝑑𝑎𝑘 1 ∑ + 𝑖𝜃(|𝑎𝑘 |2 − 1)𝑎𝑘 = −𝑎, 𝑎 = 𝑎 . (21) 𝑑𝜏 𝑁𝑒 𝑘 𝑘 The normalized radiation power and electron energy are given by formulas [38] 𝑃 = 2|𝑎|2

(22) 4. Autophasing time

and 𝐸=

1 ∑ |𝑎 |2 , 𝑁𝑒 𝑘 𝑘

In the previous section we have shown that fluctuations in the cooperative cyclotron radiation caused by shot noise depend on the degree to which the electron-oscillators are premodulated. It has been found, in particular, that premodulation results in that the relative RMSD of the autophasing time 𝛿𝑇 becomes square-root dependent on

(23)

respectively. In the absence of the energy spread, the absolute values of the amplitudes |𝑎𝑘 (0)| equal unity at the initial time and the phases 𝜙𝑘 (0) = arg(𝑎𝑘 (0)) are uniformly distributed in the interval [0; 2𝜋). 79

S.V. Anishchenko, V.G. Baryshevsky

Nuclear Inst. and Methods in Physics Research, A 879 (2018) 77–83

Fig. 1. Peak radiated power (a) and its RMSD (b) versus the nonisochronism parameter 𝜃 [black curve — 𝑁𝑒 = 6.75 ⋅ 104 and dashed curve — 1.08 ⋅ 106 ].

Fig. 2. Autophasing time (a) and its RMSD (b) versus the nonisochronism parameter 𝜃 [black curve — 𝑁𝑒 = 6.75 ⋅ 104 and dashed curve — 1.08 ⋅ 106 ].

√ 𝑁𝑒 (𝛿𝑇 ∼ 1∕ 𝑁𝑒 ) instead of logarithmically dependent (𝛿𝑇 ∼ 1∕ ln 𝑁𝑒 ). Let us show how this important result can be obtained analytically. In the absence of premodulation, the contribution to the average oscillation amplitude 𝑎𝑖𝑛 = 𝑎(0), which comes from a single particle, reads 𝑎𝑘 (0) = 𝑒𝑖𝜙𝑘 ∕𝑁𝑒 (see (21)). The contribution coming from a premodulated particle 𝑎𝑘 (0) = 𝑒𝑖𝜙𝑘 +𝑖𝛿𝜙 cos(𝜙𝑘 ) (the initial phases 𝜙𝑘 are uniformly distributed over the interval [0; 2𝜋)). By averaging over 𝜙𝑘 , we shall find the average value of 𝑎𝑖𝑛 as well as the RMSD thereof at 𝑁𝑒 ≫ 1 and 𝛿𝜙 ≪ 1: ⟨Im𝑎𝑖𝑛 ⟩ = 𝐽1 (𝛿𝜙 ), ⟨Re𝑎𝑖𝑛 ⟩ = 0, √ |⟨Im𝑎𝑖𝑛 ⟩2 − ⟨Im2 𝑎𝑖𝑛 ⟩|

1∕2

= ≈ √ √

|⟨Re𝑎𝑖𝑛 ⟩2 − ⟨Re2 𝑎𝑖𝑛 ⟩|

1∕2

= ≈ √

1 (1 + 𝐽2 (2𝛿𝜙 )) 2𝑁𝑒 1

Fig. 3. RMS fluctuations in autophasing time versus the nonisochronism parameter 𝜃 at 𝛿𝜙 = 0.05 [black curve — 𝑁𝑒 = 6.75 ⋅ 104 and dashed curve — 1.08 ⋅ 106 ].

,

2𝑁𝑒 leading to that the radiation power distribution 𝑃𝑖𝑛 = 2|𝑎𝑖𝑛 |2 at the initial time has the form: √ 𝑓 (𝑃𝑖𝑛 ) = 𝑁𝑒 𝑒−𝑁𝑒 𝑃𝑖𝑛 ∕2−𝑁𝑒 𝛼 𝐼0 (𝑁𝑒 2𝑃𝑖𝑛 𝛼). (28)

1 (1 − 𝐽2 (2𝛿𝜙 )) 2𝑁𝑒 1

,

(26)

2𝑁𝑒

where 𝐼0 is the modified Bessel function, 𝛼 = 𝐽12 (𝛿𝜙 ) ≈ 𝛿𝜙2 ∕4. Let us estimate the autophasing time by formula

where 𝐽2 (2𝛿𝜙 ) is the Bessel function. Since 𝑁𝑒 ≫ 1, then in view of the central limiting theorem, Re𝑎𝑖𝑛 and Im𝑎𝑖𝑛 have a Gaussian distribution: 𝑓 (Re𝑎𝑖𝑛 , Im𝑎𝑖𝑛 ) =

𝑁𝑒 −𝑁(Re2 𝑎 +Im2 𝑎 ) 𝑖𝑛 𝑖𝑛 , 𝑒 𝜋

𝑇 = 𝜏0 ln(𝑃0 ∕𝑃𝑖𝑛 ) = 𝜏0 ln(2∕𝑃𝑖𝑛 ). (29) √ Here, 𝜏0 = 1∕Re(−1 + 1 + 4𝑖𝜃) [38] is the time period needed for the radiation power to increase by a factor of 𝑒 in the linear instability

(27) 80

S.V. Anishchenko, V.G. Baryshevsky

Nuclear Inst. and Methods in Physics Research, A 879 (2018) 77–83

Fig. 4. Autophasing time distribution functions at 𝜏0 = 1 and 𝑁𝑒 = 6.75 ⋅ 104 [solid curve — 𝛿𝜙 = 0.05, dashed curve — 𝛿𝜙 = 0.01, dot-dashed curve — 𝛿𝜙 = 0].

Fig. 6. Radiation pulse on a normal scale (a) and on a logarithmic scale (b) [𝑁𝑒 = 1010 , 𝜃 = 10, and 𝛿𝜙 = 0].

𝐹𝑥 (𝑦) =

Fig. 5. RMSD as a function of the number of particles at 𝛿𝜙 = 0.05 [solid curve corresponds to (30), dashed curve corresponds to the asymptotic expansion at 𝛿𝜙2 𝑁𝑒 ≫ 1, dot-dashed

. (32) 𝜕𝑥2 Here, 𝛾𝑒 = 0.577 is the Euler constant and 𝐿𝑥 (𝑦) is the Laguerre polynomial. If 𝑁𝑒 𝛼 ≪ 1, then (32) transforms to the form 𝐺𝑥 (𝑦) =

curve — to the asymptotic expansion at 𝛿𝜙2 𝑁𝑒 ≪ 1].

stage. Eq. (29) implies, first, that the linear stage is much longer than the time during which nonlinear effects are essential, and, second, the radiated power in the saturation stage is 𝑃0 = 2|𝑎|2 ∼ 2 (see Fig. 6). The latter condition means coherent summation of oscillations from all the particles. Using (28) and (29), we can find the distribution function 𝑔(𝑇 ) related to 𝑓 (𝑃𝑖𝑛 ) by 𝑔(𝑇 ) = 𝑓 (𝑃𝑖𝑛 )|𝑑𝑃𝑖𝑛 ∕𝑑𝑇 |.

𝛿𝑇 = √

,

(33)

(34)

The Fig. 5 plots 𝛿𝑇 versus lg 𝑁𝑒 , respectively. It is obvious from Fig. 5 that when the number of particles exceeds a certain value depending on the degree of premodulation, the logarithmic 𝑁𝑒 dependence of the relative RMSD of the autophasing √ time 𝛿𝑇 (𝛿𝑇 ∼ 1∕ ln 𝑁𝑒 ) goes to a square-root dependence (𝛿 ∼ 1∕ 𝑁𝑒 ). As follows from (32), the value 𝑇 √ of 𝛿𝑇 𝑁𝑒 = 7.67 at 𝛿𝜙 = 0.05 and 0 < lg 𝜃 < 2 agree with the simulated ones (Fig. 3) within 20% accuracy. Let us note a very important circumstance. The duration of the cooperative cyclotron radiation pulse from nonisochronous electronoscillators is ∼ 𝜏0 [38]. The RMSD of the autophasing time in the absence of premodulation has the same order of magnitude: √ √ 𝛥𝑇 = |⟨𝑇 2 ⟩ − ⟨𝑇 ⟩2 | ≈ 𝜋𝜏0 ∕ 6 (35)

(30)

√ 𝑁𝑒 𝐼0 (2𝑁𝑒 𝛼𝑒−𝑇 ∕𝜏0 ) 𝜏0 × exp(−𝑁𝑒 𝑒−𝑇 ∕𝜏0 − 𝑁𝑒 𝛼 − 𝑇 ∕𝜏0 ).

𝜋 6(𝛾𝑒 + ln 𝑁𝑒 )

in the opposite case (𝑁𝑒 𝛼 ≫ 1), √ 2 𝛿𝑇 = √ . 𝑁𝑒 𝛼 ln(1∕𝛼)

Thus we have 𝑔(𝑇 ) =

𝜕𝐿𝑥 (𝑦) , 𝜕𝑥 2 𝜕 𝐿𝑥 (𝑦)

(31)

From the distribution functions 𝑔(𝑇 ) plotted in Fig. 4 follows that with growing premodulation, the autophasing time and its RMSD thereof decrease. Using the distribution function 𝑔(𝑇 ), we shall compute the relative RMSD 𝛿𝑇 : √ |⟨𝑇 2 ⟩ − ⟨𝑇 ⟩2 | 𝛿𝑇 = ⟨𝑇 ⟩ √ 2 (𝑁 𝛼) + 𝑒−𝑁𝑒 𝛼 𝐺 (𝑁 𝛼) 2 𝜋 ∕6 − 𝑒−2𝑁𝑒 𝛼 𝐹−1 𝑒 −1 𝑒 = , 𝑒𝑁𝑒 𝛼 (𝛾𝑒 + ln 𝑁𝑒 ) + 𝐹−1 (𝑁𝑒 𝛼)

at 𝑁𝑒 𝛼 ≪ 1. However, phase premodulation of particles leads to an appreciable decrease in the autophasing time spread: √ 𝛥𝑇 ≈ 𝜏0 2∕𝑁𝑒 𝛼 at 𝑁𝑒 𝛼 ≫ 1. 81

(36)

S.V. Anishchenko, V.G. Baryshevsky

Nuclear Inst. and Methods in Physics Research, A 879 (2018) 77–83

Fig. 7. Radiated power as a function of time at 𝜃 = 10 and different 𝛿𝑎 [solid curve — 𝛿𝑎 = 0.0, dashed curve — 𝛿𝑎 = 0.04].

Fig. 8. Peak radiated power versus the nonisochronism parameter 𝜃 at different 𝛿𝑎 [black curve — 𝛿𝑎 = 0.00, dashed curve — 𝛿𝑎 = 0.02, and dot-dashed curve — 𝛿𝑎 = 0.04].

In coherent summation of cooperative-radiation pulses, the oscillation phases of all short-pulse generators must differ by 𝛥𝜙 ≪ 𝜋, thus posing the following limitation on the average statistical spread of autophasing time 𝛥𝑡 = 𝑇𝑈 𝛥𝑇 :

It has also been demonstrated that a slight energy spread (∼ 4%) results in a twofold drop of the maximum attainable power of cooperative cyclotron radiation. The analysis made here indicates that shot noise, and electron velocity spread pose considerable constraints on the output characteristics of short-pulse CRMs and limits the possibility of coherent summation of cooperative radiation pulses from several sources.

(37)

𝛺𝑇𝑈 𝛥𝑇 ≪ 𝜋, giving, after the substitution of 𝑇𝑈 (19) and 𝛥𝑇 (36) √ √ 4 2 𝑐2 𝜃 𝑁𝑒 𝛿𝜙 ≫ . √ 𝜋 Re(−1 + 1 + 4𝑖𝜃) 𝑣2

References

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⟂0

[1] R. Bonifacio, et al., Riv. Nuovo Cimento 13 (1990) 1–69. [2] N.S. Ginzburg, et al., IEEE Trans. Plasma Sci. 41 (2013) 646–660. [3] D.I. Trubeckov, A.E. Hramov, Lectures on high-frequency electronics for physicists. V. 2, FIZMATLIT, Moscow, 2004 (in Russian). [4] S.V. Anishchenko, V.G. Baryshevsky, Nucl. Instrum. Methods B 355 (2015) 76–80. [5] V.V. Rostov, A.A. Elchaninov, I.V. Romanchenko, M.I. Yalandin, Appl. Phys. Lett. 100 (2012) 224102. [6] N.S. Ginzburg, et al., Phys. Rev. Lett. 115 (2015) 114802. [7] A.M. Kondratenko, E.L. Saldin, Part. Accel. 10 (1980) 207–216. [8] R. Bonifacio, C. Pellegrini, L. Narducci, Opt. Commun. 50 (1984) 373–378. [9] R. Bonifacio, et al., Phys. Rev. Lett. 73 (1994) 70–73. [10] E.L. Saldin, E.A. Schneidmiller, M.V. Yurkov, Opt. Commun. 148 (1998) 383–403. [11] M. Hogan, et al., Phys. Rev. Lett. 80 (1998) 289–292. [12] J. Andruszkov, et al., Phys. Rev. Lett. 85 (2000) 3825–3829. [13] M.V. Yurkov, Nucl. Instrum. Methods A 483 (2002) 51–56. [14] V.A. Atvazyan, et al., Nucl. Instrum. Methods 507 (2003) 368–372. [15] F. Lehmkuhler, et al., Sci. Rep. 4 (2014) 05234. [16] L.H. Yu, Phys. Rev. A 44 (1991) 5178–5193. [17] L.H. Yu, et al., Science 289 (2000) 932–934. [18] E. Allaria, et al., Nat. Photon. 6 (2012) 699–704. [19] J. Amann, et al., Nat. Photon. 6 (2012) 693–698. [20] G. De Ninno, et al., Nature Commun. 6 (2015) 8075. [21] N.S. Ginzburg, I.V. Zotova, A.S. Sergeev, JETP Lett. 60 (1994) 513–517. [22] N.S. Ginzburg, et al., JETP Lett. 63 (1996) 331–335. [23] N.S. Ginzburg, et al., Phys. Rev. Lett. 78 (1997) 2365–2368. [24] N.S. Ginzburg, et al., Nucl. Instrum. Methods A. 393 (1997) 352–355. [25] B.W.J. McNeil, G.R.M. Robb, D.A. Jaroszynsky, Opt. Commun. 163 (1999) 203–207. [26] S.M. Wiggins, et al., Phys. Rev. Lett. 84 (2000) 2393–2396. [27] N.S. Ginzburg, et al., Phys. Rev. E 60 (1999) 3297–3304. [28] N.S. Ginzburg, et al., Tech. Phys. 72 (2002) 83–91. [29] A.A. Elchaninov, et. al, JETP Lett. 77 (2003) 266–269. [30] A.A. Elchaninov, et al., Laser Part. Beams 21 (2003) 187–196. [31] S.D. Korovin, et al., Phys. Rev. E 74 (2006) 016501. [32] S.P. Bugaev, E.A. Litvinov, G.A. Mesyats, D.I. Proskurovskii, Phys. Usp 18 (1975) 51–61. [33] G.A. Mesyats, Plasma Phys. Control. Fusion 47 (2005) A109–A151. [34] E.B. Abubakirov, A.P. Konjushkov, A.S. Sergeev, J. Commun. Tech. Electron. 54 (2009) 959–964. [35] A.I. Gaponov, M.I. Petelin, V.K. Yulpatov, Radiophys. Quantum Electron. 10 (1967) 749–813. [36] V.V. Zheleznyakov, V.V. Kocharovskii, Vl.V. Kocharovskii, Radiophys. Quantum Electron. 29 (1987) 830–848. [37] Yu.A. Il’inskii, N.S. Maslova, Sov. Phys.—JETP 67 (1988) 96–97.

Let 𝑣∕𝑐 ∼ 0.4 and 𝜃 = 100, then we have the following estimate of √ the required degree of premodulation: 𝑁𝑒 𝛿𝜙 ≫ 86. At 𝑁𝑒 = 109 , the parameter of premodulation 𝛿𝜙 must be greater than 3 ⋅ 10−3 . 5. Energy spread To take account of the electron energy spread 𝐸𝑘 (0) ≈ 𝑎2𝑘 ∕𝑁𝑒 , we assume the initial amplitudes 𝑎𝑘 (0) to be Gaussian random variables whose mean equals unity and the relative RMSD 𝛿𝑎 ≈ 𝛿𝐸 ∕2 (𝛿𝐸 is the relative RMSD deviation of particle energy). It should be noted the Penman–McNeil algorithm is not applicable in the presence of energy spread. Therefore instead of the number 𝑁 = 36 of simulated electrons we took the number 𝑁 = 288 ≫ 1 of particles with the initial phases uniformly distributed in the interval [0; 2𝜋). Analyzing the results of numerical experiments [40], we can see that the energy spread leads to a sharp drop in the radiated power (Fig. 7). This is well-illustrated by Fig. 8, where the growing influence of the energy spread with higher 𝜃 is seen clearly: the energy spread leads to a stronger suppression of radiation at large 𝜃, and peak radiated power and radiated energy both decrease. At 𝛿𝑎 = 0.02 (𝛿𝐸 = 0.04), the maximum attainable radiated power reduces by a factor of two. 6. Conclusion It has been shown that for the number of electrons 𝑁𝑒 ∼ 109 – 1012 , typical of modern acceleration facilities, the relative RMSD of the autophasing time from its mean is 𝛿𝑇 ≈ 1.1∕ ln 𝑁𝑒 ∼ 0.04–0.05. The fluctuations in the peak radiated power 𝛿𝑃 appear to be negligibly small (𝛿𝑃 < 10−4 ). The autophasing time distribution function depending on the number of particles 𝑁𝑒 has been obtained. When the number of particles exceeds a certain value depending on the degree of premodulation, the logarithmic dependence of the relative RMSD of the autophasing time on the number of electron-oscillators (𝛿𝑇 ∼ 1∕ ln 𝑁𝑒 ) goes to a square-root √ dependence (𝛿𝑇 ∼ 1∕ 𝑁𝑒 ). 82

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[38] L.A. Vainshtein, A.I. Kleev, Sov. Phys. Dokl. 35 (1990) 359. [39] Yu.A. Kobelev, L.A. Ostrovskii, I.A. Soustova, Sov. Phys.—JETP 72 (1991) 262–267. [40] S.V. Anishchenko, V.G. Baryshevsky, Tech. Phys. 61 (2016) 934–937.

[41] L.D. Landau, E.M. Lifshiz, The Classical Theory of Fields, Butterworth Heineman, Amsterdam, 1987. [42] C. Penman, B.W.J. McNeil, Opt. Commun. 90 (1992) 82–84.

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