ISSN 10637850, Technical Physics Letters, 2013, Vol. 39, No. 5, pp. 446–449. © Pleiades Publishing, Ltd., 2013. Original Russian Text © N.M. Ryskin, N.S. Ginsburg, I.V. Zotova, 2013, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2013, Vol. 39, No. 9, pp. 86–89. SelfSimilar Modes of Amplification and Compression of Electromagnetic Pulses in Their Interaction with Electron Flows N. M. Ryskin*, N. S. Ginsburg, and I. V. Zotova Saratov State University, Saratov, 410012 Russia Institute of Applied Physics, Russian Academy of Sciences, Nizhni Novgorod, 603950 Russia *email: RyskinNM@info.sgu.ru Received January 14, 2013 Abstract—Selfsimilar solutions describing the amplification and compression of ultrashort electromagnetic pulses in their interaction with quasistationary electron flows are constructed on the basis of analogy with the coherent amplification of short optical pulses by inverted laser media. It is shown that the effects of ampli fication of pulses with their simultaneous compression are universal and must be observed for different mech anisms of induced radiation, including the Cherenkov and cyclotron mechanisms. DOI: 10.1134/S106378501305009X Experiments that revealed the amplification and compression of a short electromagnetic pulse propa gating along a quasistationary rectilinear electron flow under the conditions of the Cherenkov mecha nism of electron–wave interaction are described in [1]. It should be noted that the amplification of elec tromagnetic pulses with simultaneous shortening of their effective duration is a fairly universal effect, which must also be observed for other interaction mechanisms, including the cyclotron (cyclotron reso nance masers) and undulator mechanisms (free elec tron lasers) [2, 3]. The above effects are obviously sim ilar to the coherent amplification of short optical pulses by inverted laser media, which are described by wellknown selfsimilar solutions [4, 5]. In fact, an analysis of the numerical simulation results [1, 3] shows that the amplification of short pulses by elec tron beams exhibits a pronounced selfsimilar behav ior. Therefore, it is relevant to search for analogs of selfsimilar solutions presented in [4, 5] for different mechanisms of electron–wave interaction. For comparison, we begin the analysis with a brief description of selfsimilar modes that occur during the propagation of an optical pulse in an inverted two level medium. The evolution of an optical pulse in an active medium is described by the wellknown Max well–Bloch equations [6] for dimensionless field amplitude a, polarization P, and population differ ence n. Under the conditions of a coherent interaction (i.e., with neglect of relaxation), substitution of vari ables P = sinΦ, n = cosΦ, and a = ∂Φ/∂τ reduces the Maxwell–Bloch equations to the sineGordon equa tion [4, 5]: ∂ ⎛ ∂Φ  + ∂Φ ⎞ = sin Φ. ⎝ ∂τ ∂Z ∂τ ⎠ (1) Solutions to (1) are of particular interest; they are a function of selfsimilar variable ξ = τ(Z – τ) and are governed by the ordinary differential equation: 2 d Φ + dΦ (2) ξ   – sin Φ = 0. 2 dξ dξ Equation (2) has a solution in the form of the socalled πpulse (unlike solitons, which are referred to as 2π pulses [4]). Since a = ∂Φ/∂τ = (Z – 2τ)∂Φ/∂ξ, the amplitude of the πpulse linearly increases during its propagation. At the same time, the duration of the π pulse linearly decreases owing to the definition of the selfsimilar variable ξ. Consider the resonance interaction of electromag netic pulses with a flow of unexcited cyclotron oscilla tors under conditions of the anomalous Doppler effect as the closest classical analog of the amplification of ultrashort optical pulses in inverted media: ω – hV 0 = – ω H , where ωH is the gyrofrequency. It is assumed that cir cularly polarized wave A+ = Ax + iAy = A(z, t)exp(iωt – ihz) is decelerated and its phase velocity is less than translational velocity of the particles V0. In this case, radiation can occur in the absence of the initial trans verse velocity of the particles. In addition, in the same way as the amplification in quantum media is described by the Bloch equations for inversion and polarization, the flow of initially unexcited electron oscillators is described by two variables (without aver 446 SELFSIMILAR MODES aging over the initial phases) for the transverse momentum components (see, e.g., [7, 8]): 2 ∂a ∂a ∂p (3)  +  = p,  + ip p = a, ∂Z ∂τ ∂Z where τ and Z are the dimensionless time and coordi nate and a and p are the normalized field amplitude and transverse momentum of the electrons. The nor malization of the variables is described in more detail elsewhere—for example, in [7, 8]. The initial trans verse momentum of the electrons is absent; i.e., p(Z = 0) = 0; the emission process is accompanied by a buildup of transverse oscillations owing to the transla tional energy of the particles. It is easy to see that Eqs. (3) are invariant with respect to the scale transformation a λ3a, p 2 2 λp, τ τ/λ , and Z – τ λ (Z – τ), where λ is an arbitrary parameter. If we assume that λ = τ , we obtain the same selfsimilar variable ξ = τ(Z – τ) as in the case of the amplification of an optical pulse in an inverted medium. In fact, the substitution of the vari ables 3/2 Vol. 39 No. 5 |P| (a) 1.5 |P| 1.0 0.4 0.5 |A| 0 −20 0 ξ 20 0 40 (b) |a| 10 5 0 1/2 a ( Z, τ ) = τ A ( ξ ), p ( Z, τ ) = τ P ( ξ ) reduces system (3) to a system of ordinary differential equations: 2 dP (4) ξ dA  + 3 A = P,  + iP P = A. dξ 2 dξ Solutions to (4) can be found numerically. To elim inate the singularity at the point ξ = 0, the initial con ditions must be selected so that the relationship P(0) = 3A(0)/2 holds (cf., [4, 5]). An example of a selfsimilar solution in the case of A(0) = 0.1 is shown in Fig. 1a. The respective distribution of the electromagnetic field amplitude in the (Z, τ) plane is represented in Fig. 1b. Since the pulse propagation trajectory asymp totically tends to the line of Z = τ at high τ values, the peak amplitude of the pulse increases proportionally to Z3/2 as the length of the interaction space increases. In addition, its duration linearly decreases. Owing to this, the energy stored in the pulse increases in propor tion to the squared length of its interaction with the electron flow. The front growth rate and pulse ampli tude increase with increasing parameter of initial con ditions A(0); however, the qualitative form of the solu tion remains unchanged. These findings are con firmed by the results of numerical simulation of input equations (3). The effects of amplification and compression of ultrashort electromagnetic pulses can also occur for more applicationsignificant mechanisms of elec tron–wave interaction based on the inertial bunching of particles [1–3]. Let us consider this process using the example of the Cherenkov mechanism of amplifi cation of an electromagnetic pulse by a rectilinear electron flow in a dielectricloaded waveguide, which was experimentally studied in [1]. Under the synchro TECHNICAL PHYSICS LETTERS |A| 0.8 447 10 8 8 6 6 τ 4 4 z 2 2 0 0 Fig. 1. (a) Selfsimilar solutions |A(ξ)| and |P(ξ)| in the case of amplification of electromagnetic pulses by a flow of unexcited classical oscillators under conditions of the anomalous Doppler effect (A(0) = 0.1) and (b) respective time–space distribution of the electromagnetic field amplitude in the (Z, τ) variables. nism conditions ω ≈ h(ω)V0 in the ultrarelativistic 2 approximation γ0  1 (where γ0 = (1 – V 0 /c2)–1/2 is the relativistic mass factor), the nonstationary amplifica tion of a short electromagnetic pulse is described by the following system of equations: 2π ∂ ∂ G  +  a =  exp [ – iθ ] dθ 0 , ∂Z ∂τ π ∫ 0 ∂ε  = – Re ( a exp [ iθ ] ), ∂Z a Z=0 = a 0 ( τ ), θ 0 ∈ [ 0, 2π ), θ ε –2 ∂θ  = ε – 1, ∂Z Z=0 Z=0 (5) = θ ( θ 0 ), = 1. Here, Z and τ are the dimensionless coordinate and time, a is the normalized wave amplitude, θ = ωt – hz is the phase of the electrons with respect to the wave, θ0 is the initial phase, G is the gain parameter propor 2013 448 RYSKIN et al. ϑ 80 |A| (a) 0.03 |J| (b) 1.5 60 |J| 0.02 40 1.0 0.01 20 0.5 0 40 20 60 80 20 0 40 ξ 60 0 80 ξ |J| 1.5 1.0 0.5 0 |a| (c) 0.2 0.1 0 6 (d) 6 4 8 τ 6 2 8 τ 6 2 4 2 4 4 Z Z 2 00 00 Fig. 2. Selfsimilar solutions (a) |A(ξ)|, (b) |J(ξ)|, and ϑ(ε, ϑ0) in the case of amplification of electromagnetic pulses under the Cherenkov interaction mechanism (G = 1, r = 0.001). Respective time–space distributions of (c) electromagnetic field amplitude and (d) highfrequency current harmonics in the (Z, τ) variables. For clarity, the domain of definition of the initial phases is lim ited to three consecutive periods of ϑ ∈ [0, 6π). tional to the electron beam current, and ε = γ/γ0 is the normalized electron energy. The normalization of the variables is described in more detail elsewhere, e.g., in [3, 8]. After substitution of the variables θ = τ – Z + θ̃ and a = ã exp[i(Z – τ)], selfsimilar variables are introduced: ã ( Z, τ ) = τA ( ξ ), θ̃ ( Z, τ ) = ϑ ( ξ ), ε = ε ( ξ )/τ, (6) where ξ = τ2(Z – τ). In this case, the following system of ordinary differential equations results from (6): 2π 2ξ dA  + A = G  exp ( – iϑ ) dϑ 0 , dξ π ∫ 0 dε  = – Re ( A exp ( – iϑ ) ), dξ ϑ ( 0 ) = ϑ 0 + r cos ϑ 0 , r  1, ϑ 0 ∈ [ 0, 2π ], ε ( 0 ) = 1, 2π G A ( 0 ) =  exp [ – iϑ ( 0 ) ] dϑ 0 . π ∫ 0 Figures 2a and 2b represent an example of a selfsim ilar solution for the case of G = 1 and r = 0.001. The respective distributions of the field amplitude | ã (Z, τ)| = 2π ∫ τ|A(ξ)| and current harmonics J = (1/π) exp ( –iϑ)dϑ0 0 (7) –2 dϑ  = ε . dξ Solutions to (7) can also be numerically found under the following boundary conditions: in the (Z, τ) plane are shown in Figs. 2c and 2d. In addition, Fig. 2b illustrates the bunching of electrons with different initial phases. It is evident that the max imum current amplitude is achieved in the region of the most compact bunch of particles. According to selfsimilar substitution (6) and the definition of the selfsimilar variable, as the length of the interaction TECHNICAL PHYSICS LETTERS Vol. 39 No. 5 2013 SELFSIMILAR MODES space increases, the field amplitude increases in pro portion to Z and the duration decreases proportionally to Z1/2. In fact, the electromagnetic pulse accumulates the energy of the different fractions of the electron flow, and its peak power, in general, can exceed the kinetic power of the beam [3, 9]. Thus, in this Letter, selfsimilar solutions describ ing the amplification and compression of electromag netic pulses under the different mechanisms of elec tron–wave interaction are constructed on the basis of the analogy with the coherent amplification of optical pulses. It should be borne in mind that, in optics, the coherent interactions (i.e., the processes the charac teristic times of which are shorter than the relaxation times) can be observed in a fairly narrow class of active media characterized by a slow phase relaxation. In classical electronics, phase relaxation results from electron–electron or electron–ion collisions; under typical experimental conditions, the respective times are significantly longer than the times of development of instabilities. Owing to this, the classical analogs of coherent interactions can be regarded as relevant methods of generation and amplification of electro magnetic pulses; in particular, this is confirmed by experiments on the generation of microwave superra diance pulses by electron bunches [9–12]. Acknowledgments. This work was supported by the Russian Foundation for Basic Research, project no. 120200541a. TECHNICAL PHYSICS LETTERS View publication stats Vol. 39 No. 5 449 REFERENCES 1. M. I. Yalandin, A. G. 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