ISSN 10637850, Technical Physics Letters, 2015, Vol. 41, No. 9, pp. 820–823. © Pleiades Publishing, Ltd., 2015. Original Russian Text © S.V. Grishin, B.S. Dmitriev, V.N. Skorokhodov, Yu.P. Sharaevskii, 2015, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2015, Vol. 41, No. 17, pp. 9–17.
Generation of Dissipative Temporal Solitons in Ring SelfOscillating Systems with Amplifier Klystrons S. V. Grishin*, B. S. Dmitriev, V. N. Skorokhodov, and Yu. P. Sharaevskii Chernyshevsky Saratov State University, Saratov, 410012 Russia *email:
[email protected] Received March 27, 2015
Abstract—Theoretical and experimental results are presented that show the possibility of forming periodic pulsed signal sequences with a profile similar to that of bright solitons in a ring selfoscillating system with an amplifier klystron. The generated pulse sequences are dissipative temporal solitons formed in the ring due to the establishment of balance both between gain and loss and between time dispersion and cubic nonlinearity of the klystron. Analytical solutions have been obtained for a generator model with a twocavity amplifier klystron and an additional cavity resonator, which confirm the possibility of forming these dissipative struc tures. DOI: 10.1134/S1063785015090059
During the last several decades, much interest has been shown in ring selfoscillating microwave systems exhibiting complex dynamics (including dynamic chaos) [1–7]. Either vacuum amplifiers [1–3] or pas sive transmission lines based on ferromagnetic films [4–7] characterized by cubic nonlinearity are used as nonlinear elements in these selfoscillating systems. It was shown recently [6] that cubic nonlinearity caused by the parametric bond of spin waves in a ferromag netic film leads (along with time dispersion formed in the ring by the amplifier klystron operating in the lin ear mode) to the generation of dissipative structures, which are analogs of temporal bright solitons. These structures are formed on the envelope of the micro wave signal after many signal bypasses around the ring due to the establishment of balance both between gain and loss and between dispersion and nonlinearity. The width of temporal solitons is much longer than the sig nal delay time in the ring [6, 7]. It follows from the results obtained in [6] that the ring selfoscillating sys tem is some cell of an “artificial medium” (with gain, loss, time dispersion, and cubic nonlinearity), the characteristic time scale of which is related to the res onant klystron frequency. For the signal circulating in the ring, the selfoscillating system is a distributed transmission line composed of an infinite number of such cells, which provides infinite propagation time of the signal in it in the case where the gain exceeds the total loss. As was shown in [7], a temporal soliton can be formed at any point of the ring and its width barely changes.
(by analogy with selfoscillating systems with nonlin ear elements in the form of ferromagnetic films) allows for generating dissipative temporal solitons. The pur pose of this study was to analyze this possibility. The selfoscillating system under study (see block diagram in Fig. 1) consists of a multiresonance active element with cubic nonlinearity; the latter is a five cavity floatingdrift amplifier klystron. The input and output of the amplifier klystron are coupled by exter nal positive feedback, which includes an additional cavity resonator. The amplitude–frequency charac teristic (AFC) of the klystron measured in the linear mode (Fig. 2) has two resonance peaks, one of which (with the largest gain) is the main peak with resonant l frequency f 01 = 2797.3 MHz and the Q factor Q '01 = 386, while the other is an additional peak with fre l quency f 02 = 2824.8 MHz. The amplifier klystron AFC broadens with an increase in inputsignal power
Along with resonant properties, the amplifier klystron has cubic nonlinearity, which is due to the rearrangement of electrons in the beam. In this case, the selfoscillating system with a nonlinear klystron 820
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Fig. 1. Block diagram of the ring selfoscillating system: (1) amplifier klystron, (2) cavity resonator, (3, 5) direc tional couplers, (4) spectrum analyzer, and (6) realtime oscilloscope.
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Fig. 2. Amplitudefrequency characteristics of the amplifier klystron (curves 1, 2) and “amplifier klystron–cavity resonator” cir cuits (curve 3) measured beyond the ring at different levels of the input signal power: Pin = –5 dBm (curve 1; linear mode) and Pin = +22.5 dBm (curves 2, 3; nonlinear mode). The results were obtained at fixed values of beam current I0 = 45.5 mA and klystron accelerating voltage U0 = 2060 V.
Pin, which is due to the decrease in the loaded Q fac tors of the klystron cavities because of the rearrange ment of electrons in the beam. In the nonlinear mode, the resonant frequency of the main peak increases to nl f 01 = f01 = 2806.5 MHz and its Q factor decreases to nl
Q 01 = 326. The additionalpeak resonant frequency nl
barely changes ( f 02 = f02 = 2824.6 MHz). As will be shown below, a decrease in the loaded Q factors of cav ities of the amplifier klystron in the nonlinear mode does not lead to the generation of temporal solitons in the selfoscillating system with one klystron. To increase the Q factor of the klystron resonant curve, we used an additional cavity resonator with fre quency f03 = 2814 MHz and a loaded Q factor Q03 = 281. Figure 2 shows the AFC of the “amplifier klystron–cavity resonator” circuit measured in the nonlinear mode. It follows from the results presented that the resonant frequency of the main peak coincides in this case with frequency f03 and its Q factor becomes nl
equal to Q kr = 426. The presence of the cavity resona nl
tor thus increases the Q factor of the main peak ( Q kr > nl
Q 01 ), near which the generation of dissipative tempo ral solitons will be observed. The generation modes are controlled by beam cur rent I0 and klystron accelerating voltage U0. The microwave signal leaves the ring using directional cou plers and is fed to the inputs of a spectrum analyzer and a realtime oscilloscope with a band width of 10 GHz for analysis and subsequent processing. TECHNICAL PHYSICS LETTERS
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Figure 3 shows the power spectra and time depen dences of the amplitude and phase of the envelope of the generated signal obtained in the presence and absence of a cavity resonator in the ring. The ampli tude and phase profiles of the envelope were calculated based on the experimental time series, which were first subjected to digital processing (that is necessary for fil tering quantization noises) and then to Hilbert trans form. As a result of mathematical processing, the microwave filling and instantaneous phase were excluded from a further consideration. Envelope phase ϕ is determined as ϕ = ψ – ωct [8], where ψ is the total phase of the Hilbertconjugate signal and ωc is the frequency corresponding to the central fre quency of the generatedsignal spectrum. It follows from the results shown in Fig 3a that the generation of dissipative temporal solitons is observed at certain values of the accelerating voltage, beam cur rent, and mean signal power Pin = +22.5 dBm at the amplifier klystron input (when the klystron operates in the nonlinear mode). In this case, the generated microwave signal has a line spectrum, the distance between spectral components of which determines selfmodulation frequency fam = 7 MHz. Central spec trum frequency fc = 2811.5 MHz is near the main peak in the AFC of the “amplifier klystron–cavity resona tor” circuit (Fig. 2). A periodic pulse sequence with a profile similar to that of a bright soliton is formed in the time domain (Fig. 3a) [9]. The repetition period of these pulses is Trep = 1/fam = 143 ns, and their width (Td ~ Trep) exceeds signal bypass time around the ring τdel = 91 ns. The soliton nature of the generated struc tures is also indicated by the timeindependence of the envelope phase inside pulses [10]. As follows from the
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Fig. 3. Power spectra (on the left) and time realizations (on the right) of the amplitude (upper realizations) and phase (lower real izations) profiles obtained in the (a) presence and (b) absence of the cavity resonator in the feedback circuit. The beam current is I0 = (a) 45.5 and (b) 45 mA. The accelerating voltage is U0 = 2060 V in both cases. The dotted line in the time realization (a) indicates the soliton solution A = asch(t), where a = 0.5189 V.
results presented in Fig. 3b, no generation of a peri odic sequence of solitonlike pulses is observed in the absence of a cavity resonator in the ring. In this case, the generation of an amplitudemodulated signal is observed at a similar level Pin. To construct the model of the ring selfoscillating system under study, we used a twocavity amplifier klystron model [2] with mismatched input and output cavities and a shortened equation of linear oscillator describing the additional cavity resonator. We will assume that delay in the selfexcited oscillator under study is due to electron propagation in the drift space between the input and output klystron cavities with velocity v0, which differs from the phase signal velocity in the feedback circuit. The signal delay in the klystron model can be described using a delay line, which is set by a partial differential equation. With allowance for the above assumptions, the proposed model of ring selfoscillating system can be written as ∂A 1 + γ 1 A 1 = χ 1 A 4 , ∂t ∂A ∂A 2 + v 0 2 + ( α 0 + jδ )A 2 = χ 2 A 1 exp ( – jδt ), ∂t ∂x (1) A2 ∂A 3 + γ 2 A 3 = – jαJ 1 ( X ) exp [ j ( ϕ 0 – θ 0 ) ] exp ( jδt ) , ∂t A2
∂A 4 + γ 3 A 4 = χ 3 A 3 , ∂t where A1, A2, A3, and A4 are the dimensionless slow complex amplitudes of the input amplifier klystron cavity, delay line, output klystron cavity, and the addi tional cavity resonator, respectively; γ1,2,3 = α1,2,3 + jΔω1,2,3; α1,2,3 are the active loss in cavities; Δω1,2,3 = ω – ω01,02,03 are the frequency detunings; ω01,02,03 are the resonant cavity frequencies; χ1 and χ3 are the exci tation coefficients of the input amplifier klystron cav ity and the cavity resonator, respectively; α = ω01KMI0 is the amplification parameter; K is the characteristic resistance of the input klystron cavity; M is the modu lation efficiency coefficient; I0 is the beam current; J1 is the firstorder Bessel function, which, taking into account only the first two terms of the series, is J1 = X(8 – X2)/16; X = Mξθ0/2 is the bunching parameter; ξ = |A2(t)|/U0; U0 is the klystron accelerating voltage; θ0 = ω01l/v0 is the unperturbed transit angle; l is the drift space length; v0 = 2eU 0 /m is the initial electron velocity; e is the elementary charge; m is the electron rest mass; ϕ0 is the signal phase at the output of the delay line; α0 is the active loss in the delay line; δ is the frequency mismatch between the central frequency of the wave packet propagating in the delay line and the
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frequency of the input klystron cavity; and χ2 is the excitation coefficient of the delay line. On the assumption that Δω1,2,3 Ⰷ α1,2,3 and δ and α0 ~ 0, we find from (1) with rapidly oscillating terms neglected that χ1 χ2 χ3 X0 (2) δ = – jα exp [ j ( ϕ 0 – θ 0 ) ], Δω 1 Δω 2 Δω 3 2 where X0 = Mθ0/(2U0). We obtain the following equa tion for A2 from (1) taking into account (2): 2
∂A ∂A 2⎞ 1 ∂ A 2 2 1 j ⎛ 2 + + D + N A 2 A 2 = 0, ⎝ ∂x v 0 ∂t ⎠ 2 ∂t 2 2 [ Δω 1 ( Δω 2 + Δω 3 ) + Δω 2 Δω 3 ] where D = is the Δω 1 Δω 2 Δω 3 v 0 N = timedispersion coefficient and 3 χ1 χ2 χ3 X jα 0 exp[j(ϕ0 – θ0)] is the nonlinearity Δω 1 Δω 2 Δω 3 v 0 16 coefficient. Equation (3) is an analog to the nonlinear Schrödinger equation, one of the stationary solutions to which can have a form of a temporal bright soliton [9]. The nonlinearity coefficient in (3) is a real value if ϕ0 – θ0 = ±π/2 ± πn (n = 0, 1, 2, …). If the condition DN > 0 is satisfied, Eq. (3) has a stationary solution in the form of a bright soliton [9]. The generation of bright solitons was experimentally observed when ω01 < ωc < ω03 < ω02 (Δω1 > 0, Δω2 < 0, Δω3 < 0). Under these conditions, it follows from (3) that N < 0 if ϕ0 – θ0 = π/2 + πn (n = 0, 2, 4, …) or ϕ0 – θ0 = –(π/2 + πn) (n = 1, 3, 5, …). To determine the frequency ranges in which D can have opposite signs, it should be assumed that D = 0. As a result, we obtain threshold frequency value fth = 2809.8 MHz, which, being in the frequency range ω01 < ωth < ω02, determines the change of the sign of D in this range. For example, D > 0 at ω01 < ω < ωth and D < 0 at ωth < ω < ω03. Thus, the gen eration of bright solitons can be observed in the fre quency range ω01 < ωth < ωc < ω03 < ω02, which is in complete agreement with the experimental results (Fig. 3a). In addition, it follows from the expression obtained for D that the increase in the number of cav ities increases the number of terms (frequency detun
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ings) in the product in the denominator of D. At small frequency detunings, the increase in their number increases the D value, which causes the generation of temporal solitons in the experiment. To conclude, we should note that the results obtained are of practical interest for developing high power sources of pulsed signals based on beam or beamplasma devices without using external modulat ing circuits. Acknowledgments. This study was performed within a state contract of the Ministry of Education and Science of the Russian Federation (project no. 2014/203) and supported in part by the Russian Foundation for Basic Research (project no. 1402 00329) and a Program of the President of the Russian Federation for Support of Leading Scientific Schools (project no. NSh828.2014.2). REFERENCES 1. Yu. V. Anisimova, G. M. Vorontsov, N. N. Zalogin, et al., Radiotekhnika, No. 2, 19 (2000). 2. B. S. Dmitriev, Yu. D. Zharkov, N. M. Ryskin, and A. M. Shigaev, Radiotekh. Elektron. (Moscow) 46 (5), 604 (2001). 3. B. S. Dmitriev, Yu. D. Zharkov, S. A. Sadovnikov, et al., Tech. Phys. Lett. 37 (11), 1082 (2011). 4. B. A. Kalinikos, N. G. Kovshikov, and C. E. Patton, Phys. Rev. Lett. 80 (19), 4301 (1998). 5. A. V. Kondrashov, A. B. Ustinov, and B. A. Kalinikos, Tech. Phys. Lett. 36 (3), 224 (2010). 6. S. V. Grishin, V. N. Skorokhodov, and Yu. P. Sharae vskii, Tech. Phys. Lett. 40 (5), 378 (2014). 7. D. V. Romanenko, S. V. Grishin, A. V. Sadovnikov, et al., IEEE Trans. Magn. 50 (11), 4006304 (2014). 8. I. S. Gonorovskii, Radio Engineering Circuits and Sig nals (Sov. radio, Moscow, 1977) [in Russian]. 9. Yu. S. Kivshar and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003; Fiz matlit, Moscow, 2005). 10. J. M. Nash, P. Kabos, R. Staudinger, and C. E. Patton, J. Appl. Phys. 83 (5), 2689 (1998).
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Translated by A. Sin’kov
