ISSN 10637850, Technical Physics Letters, 2014, Vol. 40, No. 5, pp. 378–381. © Pleiades Publishing, Ltd., 2014. Original Russian Text © S.V. Grishin, V.N. Skorokhodov, Yu.P. Sharaevskii, 2014, published in Pis’ma v Zhurnal Tekhnicheskoi Fiziki, 2014, Vol. 40, No. 9, pp. 32–40.
Generation of Dissipative Structures in SelfOscillating Ring Systems upon Parametric Interaction of Spin Waves S. V. Grishin*, V. N. Skorokhodov, and Yu. P. Sharaevskii Saratov State University, Saratov, 410012 Russia *email:
[email protected] Received November 14, 2013
Abstract—A quasiperiodic sequence of pulses in a selfoscillating ring system with a resonant amplifier and a ferromagnetic film has been generated upon threewave parametric interaction of spin waves. It is estab lished that the generated pulses are analogs of dissipative bright solitons, which are formed as a result of com petition between amplification and loss, as well as between time dispersion and nonlinearity. To describe the mechanism of formation of these structures, we have proposed a model in the form of three parametrically coupled differential equations with amplification and a differential equation of linear oscillator. Under cer tain assumptions, this model has analytical solutions in time in the form of structures with a profile similar to that of bright solitons. DOI: 10.1134/S1063785014050083
Dissipative solitons are a new paradigm, which has been developed actively during the last two decades due to new concepts in the structure formation under energy fluxes [1]. In the microwave range, dissipative solitons can be observed in selfoscillating ring systems based on ferromagnetic films [2–6]. A ferromagnetic film is a nonlinear distributed medium, in which enve lope solitons can be formed as a result of competition between spatial dispersion of magnetostatic wave (MSW) and nonlinearity of the film in the presence of fourwave processes in the latter [7]. The MSW losses in selfoscillating ring systems are compensated for due to the amplification in the system, which leads to the formation of stationary sequences of dissipative solitons, the profile of which is similar to that of bright or dark solitons [2, 3]. Threewave parametric processes (along with four wave ones) can be developed in a ferromagnetic film; these processes are responsible for the generation of periodic structures in the form of relaxation oscilla tions in a selfoscillating system [8]. These structures are formed at the MSW frequency due to the competi tion between amplification and nonlinear MSW loss related to parametric excitation of spin waves at fre quencies lower than the MSW frequency by a factor of 2. The duration of relaxation oscillations is longer than the time of signal bypass through the ring by at least an order of magnitude; these oscillations are formed only when amplification occurs in the system. As was shown in [4, 6], the use of passive resonant elements in a selfoscillating system leads to the formation of a sta tionary sequence of solitonlike pulses from relaxation oscillations. However, the phase profile in these dissi pative structures is not constant; therefore, the soli
tonlike pulses generated in [4, 6] are not analogs of bright solitons. In this Letter, we show generation of pulses that are analogs of dissipative bright solitons in a selfoscillat ing ring system with a ferromagnetic film and an active resonant element. Dissipative structures are generated under conditions of threewave parametric decay of a magnetostatic surface wave (MSSW) at a certain signal power in the ring and when the oscillation frequency is close to the resonant frequency of the active element. Pulses are generated in a selfoscillating ring sys tem, the feedback circuit of which includes a spin wave MSSW transmission line (Fig. 1). A multiple cavity floatingdrift klystron, which has the properties of active and resonant elements simultaneously, is used as an amplifier. The microwave signal is extracted from the ring using two directional couplers (DC1 and DC2) to be fed to the inputs of a spectrum analyzer and a realtime oscilloscope for analysis and subse
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Fig. 1. Block diagram of a selfoscillating ring system: (1) klystron amplifier, (2) spinwave MSSW transmission line, (3) spectrum analyzer, and (4) realtime oscilloscope.
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quent processing. The spinwave transmission line has a standard configuration of a delay line, in which two 30μmwide microstrip conductors spaced by 3 mm are used to excite MSSW reception. An iron–yttrium garnet film with saturation magnetization 4πM0 = 1680 Gs and thickness d = 40 μm serves as a ferromag netic film. External dc magnetic field H0 = 425 Oe is applied parallel to the microstrip conductors. The amplifier is a fivecavity floatingdrift klystron with a resonant frequency dependence of gain (Fig. 2). The amplitude–frequency characteristic of the klystron amplifier shown in Fig. 2 exhibits three peaks, one of which (main) has resonant frequency f0 = 2798 MHz corresponding to the central klystron frequency. In the experiment, the klystron amplifier operates under smallsignal conditions and compensates for the loss in the ring. The regimes of microwavesignal genera tion are controlled by changing accelerating klystron voltage U, which leads mainly to a change in the phase shift of the microwave signal in the ring and, corre spondingly, to tuning oscillation frequency fosc with respect to the central klystron frequency [5]. Due to this specific feature, the klystron can be used as an electronic phase shifter. Figure 3 shows the power spectra and the ampli tude and phase profiles of the generated signal as a function of U. The phase profiles were calculated based on the Hilbert transformation applied to the measured time realizations. It follows from the analy sis of the power spectra in Fig. 3 that a change in the accelerating klystron voltage leads to tuning oscilla tion frequency fosc (the carrier frequency of the multi TECHNICAL PHYSICS LETTERS
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frequency signal) with respect to the central klystron frequency. At the same time, the amplitude and phase profile of the generated dissipative structures change in the time window and the quasiperiodic sequence of pulses in the form of relaxation oscillations (Fig. 3a) is transformed into a quasiperiodic sequence of soliton like pulses (Fig. 3b). In the case of relaxation oscilla tions, the phase profile in pulses has a pronounced extremum (maximum) and is asymmetric (the phase slowly increases at the pulse front and rapidly decreases at the pulse trailing edge). In the case of soli tonlike pulses (Fig. 3b), the phase profile in pulses barely changes with time, while the amplitude profile of pulses becomes almost symmetric and similar to that of dissipative bright solitons. As can be seen in Fig. 3a, the selfmodulation fre quency of the spin waves of pulses in the form of relax ation oscillations is fam = 1/Tr = 146–170 kHz (Tr is the pulse repetition period) at Δf = fosc – f0 ≈ 3 MHz and the measured pulse width at the base level Td is almost equal to the pulse period (i.e., Td ~ Tr = 5.9– 6.8 μs). The presence of a noiselike background in the power spectrum of microwave signal is most likely due to the variability of the pulserepetition period. The pulses that are analogs of dissipative bright solitons are generated at Δf ≈ –3 MHz (Fig. 3b). In this case, a decrease in the accelerating voltage increases the sig nal power in the ring, as is evidenced by the increase in the selfmodulation frequency of the spin waves fam = 375–399 kHz. Here, the pulse width decreases to Td = 2.5–2.7 μs; however, this value is larger than the time of signal bypass through the ring by a factor of 30 (τr =
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Fig. 3. (Left column) Power spectra and (right column) time realizations of the (upper realizations) amplitude and (lower real izations) phase profiles recorded at a beam current I = 17.25 mA and accelerating voltages U = (a) 2115 and (b) 1836 V. The dotted 2
line in panel (b) (time realization) indicates the soliton solution A = asech ( t ) , where a = 0.0833 V.
83 ns). Thus, the quasiperiodic sequence of pulses with an amplitude and phase profile similar to that of dissipative bright solitons is formed at a certain signal power in the ring, when the oscillation frequency is close to the central klystron frequency. The selfoscillating system under study can be pre sented as a combination of a ring of a successively con nected ferromagnetic medium, a linear inertialess amplifier, and a linear oscillator. The ferromagnetic medium in the ring is described by the system of three parametrically coupled firstorder differential equa tions; this system was analyzed for the first time in [8]. The linear oscillator and the linear inertialess ampli fier are a simplified model of klystron amplifier oper ating under smallsignal conditions. For the proposed ring model, we have the following system of four first order differential equations: m· + νm – c 0 b 1 b 2 exp ( – jδt ) = Kz, b· 1 + α k b 1 – c 1 mb *2 exp ( jδt ) = 0, (1) ·b + α b – c mb * exp ( jδt ) = 0, 2 k 2 2 1 ·z + ζz = ξm, where m, b1, and b2 are the dimensionless slow com plex amplitudes of magnetization of MSSW and spin waves, respectively; z is the dimensionless slow com plex amplitude of linear oscillator; K is the gain; ν is
the MSSW damping constant; αk is the spinwave damping constant at frequency ωk; δ = ωosc/2 – ωk; ωosc is the oscillation frequency equal to the MSSW frequency; c0, c1, and c2 are complex coupling coeffi 2
cients, which were reported in [9]; ζ = α0 + j( ω osc – 2
ω 0 )/(2ωosc); α0 is the loss in the linear oscillator; ω0 is the intrinsic frequency of the linear oscillator; and χ is the excitation coefficient of the linear oscillator. Let the spinwave amplitudes be equal; i.e., b1 = b2 = b. In addition, we make the following substitution of variables: ˆ 0 exp ( jδt ), b = bˆ 0 exp ( jδt ), m = m (2) z = ˆz 0 exp ( jδt ), ˆ 0 , bˆ 0 , and ˆz 0 are the complex amplitudes of where m the MSSW, spin wave, and linear oscillator, respec tively. Taking into account (2), system (1) can be trans formed as follows: ˆ· 0 + ( ν + jδ )m ˆ 0 – c 0 bˆ 20 = Kzˆ0 , m ˆ· (3) ˆ 0 bˆ 0* = 0, b 0 + ( α k + jδ )bˆ 0 – c 1 m ˆz· 0 + ( ζ + jδ )zˆ0 = χm ˆ 0.
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ˆ· 0 ), the equa At the highgain operation (K ˆz 0 Ⰷ m tion for complex spinwave amplitude bˆ in the case of 0
small detunings of spin waves from halved oscillation frequency δ ~ 0, with rapidly oscillating terms in sys tem of equations (3) neglected, can be written as 2 (4) D b·· + D b· + D b + D b b = 0, 1 0
2 0
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where D1 = ν, D2 = ν(αk + ζ) – χK, D3 = αk(νζ – χK), and D4 = –c0c1ζ. Expression (4) includes cubic non linearity, which is the result of cascade coupling between two quadratic effects in the first two equations of system (3) [10], and time dispersion (the second derivative with respect to time), which is caused by the presence of linear oscillator in the ring. Provided that (5) K = ν ( α k + ζ ) and ω osc = ω 0 , χ Eq. (4) is transformed into an equation with real coef ficients, 3 (6) d b·· – d b + d b = 0, 1 0
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α k ν, and d3 = α0c0c1. Thusobtained secondorder differential equation (6) is integrable, and its solution (at t ∞, b0 = b· 0 = 0) corresponds to bright envelope solitons [11], 2ν b 0 = α k sech ( α k t ). (7) d3 Using the system of equations (3), one can derive an expression relating the MSSW and spinwave com plex amplitudes: 2 c 0 ( ζ + jδ ) ˆ 0 = (8) m bˆ 0 . ( ζ + jδ ) ( ν + jδ ) – K It follows from (8) that the magnetization ampli tude is proportional to the squared spinwave ampli tude and has also a soliton solution. Thus, bright soli tonlike pulses should be generated at ωosc = ω0 or at detunings between these frequencies that are much smaller than their eigenvalues. This can be confirmed by the results obtained at fosc ~ f0 (Fig. 3b).
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We showed that a selfoscillating ring system com posed of a ferromagnetic film (in which parametric decay of MSSW occurs), an amplifier, and a resonant element can be considered as an artificial medium with time dispersion and cubic nonlinearity. The use of an active resonant element in the ring allows one to control the ring dispersion and form dissipative para metric structures similar to bright envelope solitons. Acknowledgments. This study was supported by the Russian Foundation for Basic Research (project no. 140200577), the Presidential Program for Sup port of Leading Scientific Schools (project no. NSh 828.2014.2), and by the Ministry of Education and Science of the Russian Federation in Russia (project No. 2014/203). REFERENCES 1. N. Akhmediev and A. Ankevich, Dissipative Solitons (Fizmatlit, Moscow, 2008) [in Russian]. 2. B. A. Kalinikos, M. M. Scott, and C. T. Patton, Phys. Rev. Lett. 84 (20), 4697 (2000). 3. M. M. Scott, B. A. Kalinikos, and C. E. Patton, Appl. Phys. Lett. 78 (7), 970 (2001). 4. E. N. Beginin, S. V. Grishin, and Yu. P. Sharaevskii, JETP Lett. 88 (10), 647 (2008). 5. S. V. Grishin, B. S. Dmitriev, Yu. D. Zharkov, V. N. Skorokhodov, and Yu. P. Sharaevskii, Tech. Phys. Lett. 36 (1), 76 (2010). 6. S. V. Grishin, E. N. Beginin, Yu. P. Sharaevskii, and S. A. Nikitov, Appl. Phys. Lett. 103, 022408 (2013). 7. B. A. Kalinikos, N. G. Kovshikov, and A. N. Slavin, JETP Lett. 38 (7), 413 (1983). 8. V. E. Demidov and N. G. Kovshikov, Tech. Phys. 44, 960 (1999). 9. S. V. Grishin, Yu. P. Sharaevskii, S. A. Nikitov, and D. V. Romanenko, IEEE Trans. Magn. 49 (3), 1047 (2013). 10. Yu. S. Kivshar’ and G. P. Agrawal, Optical Solitons: From Fibers to Photonic Crystals (Academic Press, 2003; Fizmatlit, Moscow, 2005). 11. G. B. Whitham, Linear and Nonlinear Waves (Wiley, New York, 1974; Mir, Moscow, 1977).
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Translated by A. Sin’kov
