Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-25T02:12:37.331Z Has data issue: false hasContentIssue false
  • English
  • Français

Whispering gallery mode resonators in microwave physics and technologies

Published online by Cambridge University Press:  08 July 2016

Alexander Barannik ,
Nickolay Cherpak ,
Alexander Kirichenko ,
Yurii Prokopenko ,
Svetlana Vitusevich  and
Vladimir Yakovenko
Alexander Barannik*
Affiliation:
O.Ya.Usikov Institute for Radiophysics and Electronics, NAS of Ukraine, 61085 Kharkiv, Ukraine. Phone: +380577203363
Nickolay Cherpak
Affiliation:
O.Ya.Usikov Institute for Radiophysics and Electronics, NAS of Ukraine, 61085 Kharkiv, Ukraine. Phone: +380577203363
Alexander Kirichenko
Affiliation:
O.Ya.Usikov Institute for Radiophysics and Electronics, NAS of Ukraine, 61085 Kharkiv, Ukraine. Phone: +380577203363
Yurii Prokopenko
Affiliation:
O.Ya.Usikov Institute for Radiophysics and Electronics, NAS of Ukraine, 61085 Kharkiv, Ukraine. Phone: +380577203363
Svetlana Vitusevich
Affiliation:
Peter Grünberg Institute (PGI-8), Forschungszentrum Jülich, 52425 Jülich, Germany
Vladimir Yakovenko
Affiliation:
O.Ya.Usikov Institute for Radiophysics and Electronics, NAS of Ukraine, 61085 Kharkiv, Ukraine. Phone: +380577203363
*
Corresponding author:A. Barannik Email: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

We review the main results of the development of whispering gallery mode (WGM) resonators and their unique applications due to their quasi-optical functionality. Several types of advanced WGM resonators are proposed by the authors. The theoretical results are described for the resonators with an analytical solution of the electromagnetic problems. Special emphasis is given to the interaction of moving charged particles and waves of cylindrical resonators. Important aspects are described concerning the developed sapphire resonators, for which an exact solution can only be found by using specially designed computer program products. A separate section of the paper is devoted to application aspects of the WGM resonators. In particular, it describes advanced solutions for overcoming the problems of measuring the small microwave (MW) surface impedance of unconventional superconductors in the form of large-area thin films and small samples under study. In addition, a demonstration of accurate complex permittivity measurements of small volumes of lossy liquids is provided. Special emphasis is given to highly stable MW signal sources, namely Ka-band transistor-based feedback oscillator and solid-state maser WGM oscillators. Recently obtained results are presented of experimental studies of the auto-oscillatory system developed on the basis of the WGM resonator with relativistic electron beam.

Keywords

New and emerging technologies and materialsCharacterization of material parametersMicrowave measurements
Type
Tutorial and Review Paper
Information
International Journal of Microwave and Wireless Technologies , Volume 9 , Issue 4 , May 2017 , pp. 781 - 796
Copyright
Copyright © Cambridge University Press and the European Microwave Association 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

[1] Vlasov, S.N.: About whispering gallery oscillations in open resonators with dielectric rod. Radiotek. Elektron., 12 (3) (1967), 572–573 (in Russian).Google Scholar
[2] Wait, J.R.: Electromagnetic whispering gallery waves in dielectric rod. Radio Sci., 2 (9) (1967), 1005–1017.CrossRefGoogle Scholar
[3] Rayleigh, L.: The problem of whispering gallery. Phil. Mag. 20 (1910), 1001–1004.Google Scholar
[4] Okaya, A.; Barash, L.F.: The dielectric microwave resonator. Proc. IRE, 50 (10) (1961), 2081–2092.Google Scholar
[5] D'Aiello, R.V.; Prger, H.J.: Dielectric resonators for microwave applications. IEEE Trans. Microw. Theory Tech., 12 (5) (1964), 549–550.Google Scholar
[6] Carret, G.B.; Kaiser, W.; Bend, W.L.: Stimulated emission into optical whispering modes of spheres. Phys. Rev., 124 (6) (1961), 1807–1809.Google Scholar
[7] Mikaelyan, A.L.; Turkov, Yu.G.: About internal parasitic modes in open resonators with dielectric rod. Radiotek. Elektron., 11 (2) (1966), 347–348 (in Russian).Google Scholar
[8] Braginskiy, V.B.; Vyatchinin, S.G.: About high-Q disk resonators. Rep. USSR, 252 (3) (1980), 584–588 (in Russian).Google Scholar
[9] Kirichenko, A.Ya.; Prokopenko, Yu.V.; Filippov, Yu.F.; Cherpak, N.T.: Quasi-Optical Solid State Resonators, Naukova Dumka, Kiev, 2008 (in Russian).Google Scholar
[10] Cherpak, N.; Barannik, A.; Filipov, Y.; Prokopenko, Y.; Vitusevich, S.: Accurate microwave technique of surface resistance measurement of large-area HTS films using sapphire quasi-optical resonator. IEEE Trans. Appl. Supercond., 13 (2) (2003), 3570–3573.Google Scholar
[11] Prokopenko, Yu.V.; Filipov, Yu.F.: Anisotropic disk dielectric resonator with conducting end faces. Tech. Phys., 47 (6) (2002), 731–736.CrossRefGoogle Scholar
[12] Dormidontov, A.V.; Prokopenko, Yu.V.: Influence of the refractivity and temperature of the ambient medium on the eigenfrequencies of quasioptical cylindrical dielectric resonators. Radiophys. Quantum Electron., 56 (6) (2013), 385–397.Google Scholar
[13] Prokopenko, Yu.V.; Filipov, Yu.F.: Influence of the finite conductivity of the end walls on the spectrum and power characteristics of the anisotropic dielectric circular-disk resonator. Telecommun. Radio Eng., 60 (7,8,9) (2003), 23–29.CrossRefGoogle Scholar
[14] Akay, M.F.; Prokopenko, Yu.; Kharkovsky, S.: Resonance characteristics of whispering gallery modes in parallel-plates-type cylindrical dielectric resonators. Microw. Opt. Technol. Lett., 40 (2) (2004), 96–101.CrossRefGoogle Scholar
[15] Barannik, A.A.; Prokopenko, Yu.V.; Filipov, Yu.F.; Cherpak, N.T.; Korotash, I.V.: Q factor of a millimeter-wave sapphire disk resonator with conductive end plates. Tech. Phys., 48 (5) (2003), 621–625.Google Scholar
[16] Barannik, A.A.; Prokopenko, Yu.V.: Axial index of “Whispering Gallery” oscillations in disk dielectric resonator. Telecommun. Radio Eng., 57 (6,7) (2002), 73–77.CrossRefGoogle Scholar
[17] Cherpak, N.T.; Barannik, A.A.; Prokopenko, Yu.V.; Filipov, Yu.F.; Vitusevich, S.A.: Microwave properties of HTS films: measurements in millimeter wave range. Low Temp. Phys., 32 (6) (2006), 608–613.Google Scholar
[18] Cherpak, N.T.; Barannik, A.A.; Bunyaev, S.A.; Prokopenko, Yu.V.; Vitusevich, S.: Measurements of millimeter-wave surface resistance and temperature dependence of reactance of thin HTS films using quasi-optical dielectric resonator. IEEE Trans. Appl. Supercond., 15 (2) (2005), 2919–2922.Google Scholar
[19] Vitusevich, S.; Cherpak, N.; Barannik, A.; Prokopenko, Yu.: Microwave impedance characterization of large-area HTS films: novel approach. Superconduct. Sci. Technol., 17 (7) (2004), 899–903.Google Scholar
[20] Barannik, A.A.; Prokopenko, Yu.V.; Filipov, Yu.F.; Cherpak, N.T.: Finite end screens influence the spectrum of oscillations in cylindrical quasioptical dielectric resonators. Tech. Phys. Lett., 29 (7) (2003), 542–543.CrossRefGoogle Scholar
[21] Prokopenko, Yu.V.; Filippov, Yu.F.; Shipilova, I.A.: Effect of a small cylindrical inhomogeneity on the eigenfrequency of a semicylindrical dielectric resonator featuring an axially homogeneous eigenmode. Tech. Phys. Lett., 33 (9) (2007), 729–731.Google Scholar
[22] Barannik, O.A.; Prokopenko, Yu.V.; Cherpak, M.T.; Shaforost, O.M.: Microwave “anomalies” in radially two-layered quasi-optical dielectric resonators filled with a lossy liquid. Dopovidi Nat. Akad. Nauk Ukraïny (Rep. NAS Ukraine), (11) (2005), 68–72.Google Scholar
[23] Prokopenko, Yu.V.; Filippov, Yu.F.; Shipilova, I.A.: Effect of a ring layer filled with various substances on the eigenfrequency and Q-Factor of a cylindrical quasi-optical dielectric resonator. Tech. Phys. Lett., 32 (4) (2006), 296–298.Google Scholar
[24] Prokopenko, Yu.V.; Filippov, Yu.F.; Shipilova, I.A.: Radial three-layered dielectric resonator with perfect conducting end walls. Telecommun. Radio Eng., 66 (10) (2007), 23–33.Google Scholar
25 Cherpak, N.T.; Barannik, A.A.; Prokopenko, Yu.V.; Smirnova, T.A.; Filipov, Yu.F.: A new technique of dielectric characterization of liquids, in Rzoska, S.J. and Zhelezny, V.P. (eds.), Book: Nonlinear Dielectric Phenomena in Complex Liquids, NATO Science Series, Kluwer Academic Publishers, Netherlands, vol. 157 (2004), 63–76.Google Scholar
[26] Prokopenko, Yu.V.; Filippov, Yu.F.; Shipilova, I.A.: Distribution of the field of whispering-gallery modes in a radially nonuniform two-layer cylindrical dielectric resonator. Radiophys. Quantum Electron., 51 (7) (2008), 561–570.Google Scholar
[27] Barannik, A.A.; Prokopenko, Yu.V.; Smirnova, T.A.; Filipov, Yu.F.; Cherpak, N.T.: A quasi-optical dielectric ring-resonator with conducting endplates. Telecommun. Radio Eng., 60 (7) (2003), 48–54.Google Scholar
[28] Prokopenko, Yu.V.; Suvorova, O.A.; Filippov, Yu.F.; Shipilova, I.A.: Natural oscillations of radially three-layered dielectric resonators. Radioelectron. Commun. Syst., 52 (1) (2009), 7–15.Google Scholar
[29] Prokopenko, Yu.V.; Smirnova, T.A.; Filippov, Yu.F.: Eigenmodes of an anisotropic dielectric ball. Tech. Phys., 49 (4) (2004), 459–465.Google Scholar
[30] Derkach, V.N.; Filipov, Yu.F.; Plevako, A.S.; Prokopenko, Yu.V.; Smirnova, T.A.: Determination of microwave parameters of isotropic mediums by using an open quasi-optical spherical resonator. Int. J. Infrared Millim. Waves, 25 (1) (2004), 139–148.Google Scholar
[31] Litvinenko, V.S.; Prokopenko, Yu.V.; Filipov, Yu.F.: Surface electromagnetic oscillations in a semiconductor sphere. Telecommun. Radio Eng., 60 (3, 4) (2003), 24–29.Google Scholar
[32] Prokopenko, Yu.V.; Filippov, Yu.F.; Yakovenko, V.M.: Excitation of oscillations in a hemispherical dielectric resonator by a radial magnetic dipole. Tech. Phys., 50 (5), (2005), 636–641.Google Scholar
[33] Prokopenko, Yu.V.; Filippov, Yu.F.; Shipilova, I.A.; Yakovenko, V.M.: Whispering gallery modes in a hemispherical isotropic dielectric resonator with a perfectly conducting planar surface. Tech. Phys., 51 (2) (2006), 248–257.Google Scholar
[34] Cherpak, N.T.; Barannik, A.A.; Bunyaev, S.A.; Prokopenko, Yu.V.; Torokhtii, K.I.; Vitusevich, S.A.: Millimeter-wave surface impedance characterization of HTS films and single crystals using quasi-optical sapphire resonators. IEEE Trans. Appl. Supercond., 21 (3) (2011), 591–594.Google Scholar
[35] Barannik, A.A.; Bunyaev, S.A.; Cherpak, N.T.; Prokopenko, Y.; Kharchenko, A.A.; Vitusevich, S.A.: Whispering gallery mode hemisphere dielectric resonators with impedance plane. IEEE Trans. Microw. Theory Tech., 58 (10) (2010), 2682–2691.Google Scholar
[36] Kirichenko, A.Y.; Prokopenko, Y.V.; Suvorova, O.A.; Filippov, Y.F.: Radially two-layer sphere as a sensor of dielectric characteristics of a liquid into which it is submerged. Telecommun. Radio Eng., 69 (18) (2010), 1661–1672.CrossRefGoogle Scholar
[37] Dormidontov, A.V.; Prokopenko, Yu.V.; Khankina, S.I.; Yakovenko, V.M.: Energy loss of a charged particle moving along the helical path. Telecommun. Radio Eng., 73, (13) (2014), 1165–1189.CrossRefGoogle Scholar
[38] Dormidontov, A.V.; Prokopenko, Yu.V.; Khankina, S.I.; Yakovenko, V.M.: Energy loss of charged particle on the eigenmode excitation in cylindrical structures with two-dimensional electron gas. Radiophys. Electron., 5(19) (4) (2014), 63–72.Google Scholar
[39] Barannik, A.A.; Bunyaev, S.A.; Cherpak, N.T. and Vitusevich, S.A.: Quasi-optical sapphire resonators in the form of a truncated cone. J. Lightw. Technol., 26 (17) (2008), 3118–3123.CrossRefGoogle Scholar
[40] Barannik, A.; Cherpak, N.T.; Tanatar, M.A.; Vitusevich, S.; Skresanov, V.; Canfield, P.C. et al. : Millimeter-wave surface impedance of optimally-doped Ba(Fe1−xCo x )2As2 single crystals. Phys. Rev. B, 87 (1) (2013), 014506(1–7).Google Scholar
[41] Barannik, A.A.; Cherpak, N.T.; Prokopenko, Yu.V.; Filipov, Yu.F.; Shipilova, I.A.: Whispering gallery mode sapphire resonator for microwave characterization of lossy liquids, in Proc. Int. Kharkov Symp. on Physics and Engineering of Microwaves, Millimeter and Sub-Millimeter Waves (MSMW-2007), vol. 2, 2007, 922–924.Google Scholar
[42] Gubin, A.I.; Barannik, A.A.; Protsenko, I.A.; Cherpak, N.T.; Vitusevich, S.; Offenhaeusser, A.: Biochemical liquids permittivity characterization technique based on channel, in Proc. of 43rd European Microwave Conf., Nuremberg, Germany, 2013, 314–317.Google Scholar
[43] Mazierska, J.; Wilker, C.: Accuracy issues in surface resistance measurement of high temperature superconductors using dielectric resonators. IEEE Trans. Appl. Supercond., 11 (4) (2001), 4140–4147.Google Scholar
[44] Izhyk, E.V.; Kirichenko, A.Ya.; Revenko, Yu.F.; Svistunov, V.M.; Cherpak, N.T.; Yakovenko, V.M.: Superconducting YBaCuO ceramics in high and extremaly high frequency fields. Dopovidi AS UkrSSR (in Russian Reports of AS of Ukr. SSR) (series A) (5) (1989), 51–54.Google Scholar
[45] Cherpak, N.T.; Kirichenko, A.Ya.: Behaviour of quasi-optical dielectric resonatorwith high-Tc superconductorsin the temperature range 10–300 K. Cryogenics, 31 (5) (1991), 384–387.Google Scholar
[46] Cherpak, N.T.; Barannik, A.A.; Gubin, A.I.: WGM resonator-based measurement technique for weakly and highly absorbing substances, in 20th Int. Conf. on Microwaves, Radar and Wireless Communications (MIKON), 2014, 54–57.Google Scholar
[47] Cherpak, N.T.; Barannik, A.A.; Bunyaev, S.A.: Quasi-optic dielectric resonator-based technique of HTS film millimeter wave surface resistance measurements: three types of resonators, in Proc. of 38th European Microwave Conf., Amsterdam, the Netherlands, 2008, 807–811.Google Scholar
[48] Barannik, A.A.; Cherpak, N.T.; Chuyko, D.E.: Q-factor measurement of quasi-optical dielectric resonators under conditions of the whispering gallary mode degeneration removal. IEEE Trans. Instrum. Meas., 5 (1) (2006), 70–73.Google Scholar
[49] Skresanov, V.N.; Glamazdin, V.V.; Cherpak, N.T.: The novel approach to coupled mode parameters recovery from microwave resonator amplitude -frequency response, in Proc. of 41st European Microwave Conf., Manchester, UK, 2011, 826–829.Google Scholar
[50] Barannik, A.A.; Bunyaev, S.A.; Cherpak, N.T.: On the low-temperature microwave response of a YBa2Cu3O7−δ epitaxial film determined by a new measurement technique. Low Temp. Phys., 34 (12) (2008), 977–981.Google Scholar
[51] Barannik, A.A.; Cherpak, N.T.; Kharchenko, M.S.; Semerad, R.; Vitusevich, S.A.: Surface impedance of YBa2Cu3O7−δ films grown on MgO substrate as a function of film thickness. J. Supercond. Nov. Magn., 26 (1) (2013), 43–48.Google Scholar
[52] Wu, Y. et al. : Microwave properties of BaFe1.9Ni0.1As2 superconducting single crystal. J. Supercond. Nov. Magn. 26 (2013), 1221–1225.Google Scholar
[53] Cherpak, N.T.; Barannik, A.A.; Prozorov, R.; Tanatar, M.; Velichko, A.V.: On the determination ofthe quasiparticle scattering ratein unconventional superconductors by microwave surface impedance. Low Temp. Phys., 39 (12) (2013) 1110–1112.Google Scholar
[54] Barannik, A.A.; Cherpak, N.T.; Prokopenko, Yu.V.; Filipov, Yu.F.; Shaforost, E.N.; Shipilova, I.A.: Two-layered disc quasi-optical dielectric resonators: electrodynamics and application perspectives for complex permittivity measurements of lossy liquids. Meas. Sci. Technol., 18 (2007), 2231–2238.Google Scholar
[55] Gubin, A.I.; Barannik, A.A.; Cherpak, N.T.; Vitusevich, S.; Offenhaeusser, A.; Klein, N.: Whispering-gallery mode resonator technique for characterization of small volumes of biochemical liquids in microfluidic channel, in Proc. of 41st European Microwave Conf., Manchester, UK, 2011, 615–618.Google Scholar
[56] Gubin, A.I.; Lavrinovich, A.A.; Cherpak, N.T.: Dielectric resonators with “whispering-gallery” waves in investigations of small-volume binary solutions. Ukr. J. Phys., 51 (7) (2006), 723–727 (in Ukrainian).Google Scholar
[57] Gubin, A.I.; Barannik, A.A.; Cherpak, N.T.; Protsenko, I.A.; Offenhaeusser, A.; Vitusevich, S.: Whispering-gallery mode resonator technique with microfluidic channel for permittivity measurement of liquid. IEEE Trans. Microw. Theory Tech, 63 (6) (2015), 2003–2009.Google Scholar
[58] Locke, C.R.; Ivanov, E.N.; Hartnett, J.G.; Stanwix, P.L.; Tobar, M.E.: Invited article: design techniques and noise properties of ultrastable cryogenic cooled sapphire-dielectric resonator oscillators. Rev. Sci. Instrum., 79 (2008), 051301-1–051301-12.Google Scholar
[59] Braginsky, V.B.; Ilchenko, V.S.; Bagdassarov, K.S.: Experimental observation of fundamental microwave absorption in high quality dielectric crystals. Phys. Lett. A, 120 (1987), 300–305.Google Scholar
[60] Thorne, K.S.: Gravitational radiation. In Three Hundred Years of Gravitation , Hawking, S.W. and Israel, W. (eds.), University Chicago Press, Chicago, 1987, 330.Google Scholar
[61] Tobar, M.E.; Ivanov, E.N.; Locke, C.R.; Hartnet, J.G.: Difference frequency technique to achieve frequency-temperature compensation in whispering-gallery sapphire resonator-oscillator. Electron. Lett., 38 (17) (2002), 948–950.Google Scholar
[62] Santarelli, G.; Laurent, Ph.; Lemonde, P.; Clairon, A.; Mann, A.G.; Chang, S. et al. : Quantum projection noise in an atomic fountain: a high stability cesium frequency standard. Phys. Rev. Lett., 82 (1999), 4619.Google Scholar
[63] Vitusevich, S.A.; Schieber, K.; Ghosh, I.S.; Klein, N.; Spinnler, M.: Design and characterization of an all-cryogenic low phase-noise sapphire K-band oscillator for satellite communications. IEEE Trans. Microw. Theory Tech., 51 (1) (2003), 143–169.Google Scholar
[64] Leeson, D.B.: A simple model of feedback oscillator noise spectrum. Proc. IEEE, 54 (1966), 329–330.Google Scholar
[65] Le Floch, J.M. et al. : Invited Article: dielectric material characterization techniques and design of high-Q resonators for applications from micro to millimeter-waves frequencies applicable at room and cryogenic temperatures. Rev. Sci. Instrum., 85 (2014), 031301-1–031301-13.Google Scholar
[66] Tobar, M.E.; Ivanov, E.N.; Hartnett, J.G.; Cros, D.: High-Q frequency stable dual-mode whispering gallery sapphire resonator, in IEEE MTT-S Int. Microwave Symp., Phoenix, Arizona, 20–25 May 2001, 205–208.Google Scholar
[67] Tutt, M.N.; Pavlidis, D.; Khatibzadeth, A.; Bayraktaroglu, B.: Investigation of HBT oscillator noise through 1/f noise and noise upconversion studies, in IEEE MTT-S Int. Microwave Symp. Digest, 1992, 727–730.Google Scholar
[68] Danylyuk, S.V. et al. : Phase noise study of AlGaN/GaN HEMT X-band oscillator. Phys. Status Solidi (c), 2 (7) (2005), 2615–2618.Google Scholar
[69] Oxborrow, M.: Whispering gallery oscillator (using paramagnetic ions), UK Patent GB 2425221A and US Patent 7292112, 2005–2009.Google Scholar
[70] Bourgeois, P.-Y.; Bazin, N., Giordano, V., Tobar, M.E., Oxborrow, M.: Maser oscillation in a whispering-gallery-mode microwave resonator. Appl. Phys. Lett., 87 (2005), 224104-3–224104-3.Google Scholar
[71] Greedom, D. et al. : High-power solid-state sapphire whispering gallery mode maser, in Proc. of Joint FCS (2009) and 23rd EFTF, Besancon, France, 2009, 282–285.Google Scholar
[72] Siegman, A.E.: Microwave Solid-State Masers, McGraw-Hill Book Company, New York, Toronto, London, San Francisco, 1964.Google Scholar
[73] Cherpak, N.T.: Quantum Paramagnetic Amplifiers (Solid-State Masers) of a Distributed Type in mm-Wave Range. Naukova Dumka, Kiev, 1996 (in Russian).Google Scholar
[74] Benmessaї, K.; Creedon, D.; Tobar, M.E.; Bourgeois, P.-Y.; Kersale, Y.; Giordano, V.: Measurement of fundamental thermal noise limit in a cryogenic sapphire frequency standard using bimodal maser oscillations. arXiv: 0807.2582, 2008.Google Scholar
[75] Kirichenko, A.Ya.; Cherpak, N.T.: On solid-state stable quantum oscillator in millimeter wavelength range, in Tverdotel'nyje i preobrazovatel'nyje pribory mm i submm diapazonov voln (in Russian solid-state oscillating and converting devices of mm and submm wavelength range), Kharkiv, IRE AS Ukr. SSR, 1989, 149–150.Google Scholar
[76] Galaydych, K.V.; Lonin, Yu.F.; Ponomarev, A.G.; Prokopenko, Yu.V.; Sotnikov, G.V.: Mathematical model of an excitation by electron beam of “whispering gallery” modes in cylindrical dielectric resonator. Problems of Atomic Science and Technology, Series, Plasma Phys., 16 (6) (2010), 123–125.Google Scholar
[77] Galaydych, K.V.; Lonin, Yu.F.; Ponomarev, A.G.; Prokopenko, Yu.V.; Sotnikov, G.V.: Nonlinear analysis of mm waves excitation by high-current REB in dielectric resonator. Problems of Atomic Science and Technology, Series, Plasma Phys., 18 (6(82)) (2012), 158–160.Google Scholar
[78] Dormidontov, A.V. et al. : Auto-oscillatory system based on dielectric resonator with whispering-gallery modes. Tech. Phys. Lett., 38 (1) (2012), 85–88.Google Scholar
[79] Parker, R.K.; Abrams, R.H.; Danly, B.G.; Levush, B.: Vacuum electronics. IEEE Trans. Microw. Theory Tech., 50 (3) (2002), 835–845.Google Scholar