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Mode-locked free-electron laser oscillator
NUCLEAR INSTRUMENTS
Nuclear Instruments and Methods in Physics Research A318 (1992) 114-116 North-Holland
S IN PHYSICS RESEARCH Section A
ode-locked free-electron laser oscillator Eli Jerby ', George Bekefi and Toru Hara 2
Department of Piarsics, Research Laboratory of Electronics, and Plasma Fusion Center, Massachusetts Institute Cambridge. 8114 02139. USA
of
Technology,
In this experiment we observe omission of a single RF macropulse from a mode-locked FEL oscillator. The FEL employs a continuous electron beam and operates in the microwave regime . The amplitude mode-locking is achieved by modulating the attenuation of the FEL ring cavity at its round-trip frequency ( - 27 MHz). The observed RF macropulse consists of short RF micropulses. Their period ( - 37 ns) is determined by the modulating frequency, and their width is comparable to the slippage time ( - 3 ns). Examples of experimental results of amplitude mode-locked FEL signals are presented for coaxial-line and waveguide cavity sections. Mode-locked oscillators have been widely studied in microwaves and conventional lasers . In the early i0s, Cutler demonstrated a regenerative pulse generator [ 1] by mode-locking of a microwave oscillator. He obtained - 3 ns micropulses at 4 GHz frequency from an oscillator consisting of a traveling-wave amplifier, a ring waveguide cavity, and an expander (in the form of a double-balanced mixer). More recently, various m e-locking effects were studied in the microwave regime with solid-state devices [2]. In conventional (atomic and molecular) lasers, mode-locking effects have been subjects of interest for many years [3] . Different mechanisms of mode-locking have been studied, such as self (spontaneous) mode-locking [4,5] AM and FM mode-locking [6], injection mode-locking [7], and soliton-lasers [S] . In FEL oscillators, radiation bursts and spikes have been observed in the nonlinear regime by several groups [9-13]. The appearance of bursts in the nonlinear FEL regime comes about as a re,uii of the FEL sideband instability (caused by electron oscillations in the potential wells of the ponderomotive wave). Self mode-locking operation of an FEL oscillator with a continuous electron beam has been reported recently [14,15] . Injection locking of an FEL was studied in order to achieve a single mode continuous operation [161. Loss modulation of an FEL cavity was demonstrated [17] using a cadmium telluride electrooptic cell as a means to control the number of micropulses in the FEL macropulse signal . Faculty of Engineering, Tel Aviv University, Ramat Aviv, 69978, Israel . - Department of Nuclear Engineering, Tokyo University, Tokyo 113, Japan. ()168-9(N)2/92/$(15 .11() ,0
In a previous experiment [1H] we observed short electromagnetic radiation bursts in the start-up phase of an FEL oscillator . These occured well before saturation and near the oscillation threshold where lineal phenomena dominate the interaction . The observe radiation bursts consisted of periodic micropulses contained within a bell-shaped macropulse envelope . The start-up of the radiation macropulses was found to b( correlated with random current spikes superposed on i uniform current density beam. These experimental ob nervations agreed with a theoretical linear model of the FEL impulse response in the time domain [19]. In the experiment presented here, the FEL oscilla for is mode-locked by modulating its cavity losses . As result of this modulation, the FEL produces a single macropulse of RF radiation rather than random, par tially overlapping, macropulses . A scheme of the mode-locked FEL oscillator i shown in fig . 1 . The accelerating potential is suppliei by a Marx generator (Physics International Pulsera 615 MR) . The continuous electron beam is generate by a thermionically emitting, electrostatically focuse( Pierce-type electron gun (250 kV, 250 A) from a SLAB it- 3.2 m (wovegu i de or coox ) --~ To osc+lloscol
WMAVA
PIN
WIGGLER
solenoid To MARX power suppy
w~~woo~~~~
0
27 MHz modulatic Isolotor
WVAVd 'VAVAVAVAVAVAV/ ThermiornC Pierce gun -2.25m-
Fig . l . The mode-locked FEL oscillator experimental setup
1992 - Elsevier Science Publishers B.V. All rights reserved
E. Jerby et al. / Mode-locked FEL oscillator
klystron (Model 343). An emittance selector is used to limit the beam current to - 1 A. The actual electron beam energy in this experiment is - 150 kcV, and the voltage droop is - 1 kV/p.s. An assembly of focusing coils transports the electron beam into the rectangular stainless steel drift tube (0.40 in. x 0.90 in.), which also acts as the wavcguidc for the electromagnetic radiation . The beam is contained by a uniform 1.6 kG axial magnetic field produced by a solenoid. The 65-period circularly-polarized magnetic wiggler is generated by bifilar conductors [20]. It has a period of lw = 3.5 cm and an amplitude of BW = 200-400 G. Since an aperture limits the size of the electron beam to r,, = 0.071,,., the wiggler field appears nearly sinusoidal to the drifting electrons. At the wiggler entrance a slowly increasing field amplitude is produced by resistively loading the first six periods of the wiggler magnet . The 2.7-m-long drift tube acts as a rectangular wavcguidc whose fundamental TIE,, mode has a cutoff frequency of 6.6 GHz . The system is operated in a frequency range between 8 and 11 GHz. At those frequencies the empty wavcguidc can support only the fundamental (TE ,) mode, all higher modes being evanescent . The output of the FEL is injected back to its input by a feedback line which forms a ring cavity loop. The perimeter of the ring cavity is 7.6 m. One section of the ring (3.2 m) is the feedback line, which can be either a nondispersive coaxial line (RG-214), or a waveguide section (WR-90). The modulation of the feedback attenuation is done by a PIN diode modulator (M.A. 8319-1 X23). The loss of the ring cavity (without the modulation) is 10 dB for a coaxial line, or 5.5 dB for a waveguide feedback section . The single-pass FEL gain varies so that the overall system net-gain is less than 3 dB. It is in this low net-gain operating regime that all of our measurements are carried out, and where the periodic RF macropulses are the clearest. In order to observe them, the radiation field of the ring cavity is sampled by means of a 17.2 dB directional-coupler and then measured with a crystal detector (HP423A) calibrated to 0.12 mW/mV at 100 mV. The signals are recorded by a digital oscilloscope (HP 54516A). This experimental setup resembles that of our previous experiment [18,19] . The main changes in the cavity are the installation of a PIN diode variable attenuator (modulated in 27 MHz), the removal of the RF filters, and the optional operation with a coaxial line in the feedback section . Another major modification in the present experiment is the reduction of the Marx voltage droop from 5 kV/Rs to 1 kV/Ws. The mode-locked FEL oscillator emits a clear single macropulse of RF radiation in each shot . The observed signals resemble the FEL irnpi.lse response computed
115
0.08
u
0.06
a ô 0.04 0
û
0.02
0.00 li 200
300
400
500
600
700
800
900
Fig . 2. A typical macropulse output signal of a mode-locked FEL with a coaxial line ring cavity. in ref. [ 19]. Fig . 2 shows an example of the FEL output with a coaxial feedback line. A single clear macropulse is observed with a period of - 37 ns and a micropulse width of - 3.5 ns. Taking into account the detector nonlinear response, the width of the RF micropulse is estimated to be - 2.8 ns. Fig . 3 shows a typical macropulse with a waveguide section as a feedback loop. The micropulse width in this case is twice as wide as with the coaxial line, as expected due to the waveguide dispersion. In the experimental results shown in fig. 2, for a coaxial feedback, line. the micropulse width (- 2.8 ns) 0.07 0.06 ,-, 0.05
u
a 0.04 0 U
v
v0
0_03 0.02 0.01 0.00 500
V~UÛÛÜÜ 1500
1000 t
2000
[r,S ]
Fig. 3 . A typical output of a mode-locked FEL with a waveguide ring cavity. 111. FEL EXPERIMENTS
E. Jerhy et al. / Moule-lockeol FEL oscillator
is close to the slippage time ( - 2.5 ns) computed in ref. [19]. The FEL response to a current impulse (namely, its temporal Greens function) is a finite pulse of radiation at the FEL resonance frequency. Its pulse width is
where t°- is the axial electron beam velocity, rg is the radiation group velocity, and L , is the wiggler length . The pulse width r,, known as the FEL slippage time (1), is the difference between the propagation times of an electron and a wave packet traveling along the wiggler axis. In the FEL oscillator, this pulse is circulating in the cavity and reamplified in each round-trip. As shown in ref. [191, this pulse tends to preserve its width, and its widening due to the waveguide dispersion is balanced by the FEL gain and phase shift. The
mode-locking
mechanism
enables
the
mi-
within short time-n-iaulows synchronized with the micropuls,: round-trip frequency. This time dependence of the loop net-gain results in a single clear macropulse, unlike our previous experiments [18.191 . in which we observed partially overlapping macropulses. In the frequency domain . the longitudinal modes under the FEL gain curve (whose cropulse train to evolve only
bandwidth is - 1/-.,) are locked : thus. their composition in the time domain results in narrow pulses with a macropulse width limited by the slippage time .
The clarity of the experimental results and the
theoretical analysis leads us to conclude that the effect observed has a general validity, and that it can be applicable: to other FEL oscillators, and in particular those with continuous electron beams . The electrostatic accelerator FEL [161 . for instance, has a special importance because of its unique CW operating mode and its potential spectral purity . Implementation of a mode-locking mechanism in such an FEL may extend its capabilities with a feature of macropulse operation, in addition to its CW mode . The width of the modelocked micropulses can reach. i n principle, the slippage time (1) which is given in the relativistic limit by ®, = L,,l2y_ c . where c is the speed of light, and y- _ [I - (t- ./C)']-' ' 2 is an axial relativistic factor . With the parameters of a typical electrostatic accelerator FEL (L,, = 3 m, y_ = 10) the expected macropulse width
is 50 ps. Much shorter pulses can be obtained with shorter wigglers and/or higher electron energies. In conclusion, the mode-locked FEL oscillator demonstrated in this experiment reveals aspects of the FEL Physics in the time domain . In practice, it may lead to applications at shorter wavelength, such as a two-mode (CW/short-pulse) FEL .
References [1] C .C. Cutler, Proc. IRE 43 (1955) 140 . [?] L.A. Glasser and H .A . Haus, IEEE Trans . Microwave Theory and Tech . 26 (1978) 62 . [31 P.W . Smith, Proc. IEEE 58 (1970) 1342, for recent studies on ultrafast laser phenomena . see the special issue of IEEE J . Quantum Electron . 25 (1989), [4] P .W. Smith . IEEE J . Quantum Electron . 3 (1967) 627. [5j J .R. Fontana. IEEE J. Quantum Electron . 8 (1972) 699 . [6] D.J . Kuizenga and A.E . Siegman, IEEE J . Quantum Electron . 6 (1970) 694. [7] S. Basu and R .L . Byer. IEEE J . Quantum Electron. 26 (1990) 149. [8] L .F. Mollenauer and R .H . Stolen . Opt. Let( . 9 (1984) 13 . [9] R .W. Warren. B.E . Newna m and J .C . Goldstein, IEEE J . Quantum Electron . 21 (1985) 882 . [10] J .C. Goldstein, B .W. Newnam . R.W. Warre n and R.L. Sheffield, Nucl . Instr . and Meth . A250 (1986) 4 . [11] J . Masud. T .C. Marshall, S .P. Schlesinger and F .G . Yee . Phys . Rev. Lett . 5 6 (1986) 1567 . [12] J .W. Dodd and T .C . Marshall, Nucl . Instr. and Meth . A296 (1990) 4. [ 13] B .A . Richman . J .M .J . Made y and E . Szarmes, Phys . Rev . Lett . 63 (1989) 1682 . 1141 T . Kawamura . K. Toyoda and M . Kawai . Appl . Phys . Lett . 5 1 (1987) 795 . [151 Y . Kawamura . B .C. Lee. M . Kawa i and K . Toyoda, Phys . Rev . Lett . 67 (1991) 832 . [16] A . Amir, R .J . Ho, F . Kielmann, J . Mertz and L .R. Elias, Nucl . Instr. and Meth . A272 (1988) 174 . [17] S .V . Benson, J .M .J . Madey, E .B . Szarmes, A . Bhowmik, P . Metty and M . Curtin, Nucl . Instr. and Meth . A296 (19911) 762 . [181 E . Jerby, G . Bekefi and J . Wurtele, Phvs . Rev . Lett . 66 (1991) 2068, also Nucl . Instr. and Meth . A304 (1991) 1(17 . [191 E . Jerby, G . Bekefi and J . Wurtele, IEEE J . Quantum Electron . 27 (1991) 2512 . [20] J . Fajans, G . Bekefi, Y .Z . Yin and B. Lax, Phys . Fluids 28 (1985) 1995 .
Nuclear Instruments and Methods in Physics Research A318 (1992) 114-116 North-Holland
S IN PHYSICS RESEARCH Section A
ode-locked free-electron laser oscillator Eli Jerby ', George Bekefi and Toru Hara 2
Department of Piarsics, Research Laboratory of Electronics, and Plasma Fusion Center, Massachusetts Institute Cambridge. 8114 02139. USA
of
Technology,
In this experiment we observe omission of a single RF macropulse from a mode-locked FEL oscillator. The FEL employs a continuous electron beam and operates in the microwave regime . The amplitude mode-locking is achieved by modulating the attenuation of the FEL ring cavity at its round-trip frequency ( - 27 MHz). The observed RF macropulse consists of short RF micropulses. Their period ( - 37 ns) is determined by the modulating frequency, and their width is comparable to the slippage time ( - 3 ns). Examples of experimental results of amplitude mode-locked FEL signals are presented for coaxial-line and waveguide cavity sections. Mode-locked oscillators have been widely studied in microwaves and conventional lasers . In the early i0s, Cutler demonstrated a regenerative pulse generator [ 1] by mode-locking of a microwave oscillator. He obtained - 3 ns micropulses at 4 GHz frequency from an oscillator consisting of a traveling-wave amplifier, a ring waveguide cavity, and an expander (in the form of a double-balanced mixer). More recently, various m e-locking effects were studied in the microwave regime with solid-state devices [2]. In conventional (atomic and molecular) lasers, mode-locking effects have been subjects of interest for many years [3] . Different mechanisms of mode-locking have been studied, such as self (spontaneous) mode-locking [4,5] AM and FM mode-locking [6], injection mode-locking [7], and soliton-lasers [S] . In FEL oscillators, radiation bursts and spikes have been observed in the nonlinear regime by several groups [9-13]. The appearance of bursts in the nonlinear FEL regime comes about as a re,uii of the FEL sideband instability (caused by electron oscillations in the potential wells of the ponderomotive wave). Self mode-locking operation of an FEL oscillator with a continuous electron beam has been reported recently [14,15] . Injection locking of an FEL was studied in order to achieve a single mode continuous operation [161. Loss modulation of an FEL cavity was demonstrated [17] using a cadmium telluride electrooptic cell as a means to control the number of micropulses in the FEL macropulse signal . Faculty of Engineering, Tel Aviv University, Ramat Aviv, 69978, Israel . - Department of Nuclear Engineering, Tokyo University, Tokyo 113, Japan. ()168-9(N)2/92/$(15 .11() ,0
In a previous experiment [1H] we observed short electromagnetic radiation bursts in the start-up phase of an FEL oscillator . These occured well before saturation and near the oscillation threshold where lineal phenomena dominate the interaction . The observe radiation bursts consisted of periodic micropulses contained within a bell-shaped macropulse envelope . The start-up of the radiation macropulses was found to b( correlated with random current spikes superposed on i uniform current density beam. These experimental ob nervations agreed with a theoretical linear model of the FEL impulse response in the time domain [19]. In the experiment presented here, the FEL oscilla for is mode-locked by modulating its cavity losses . As result of this modulation, the FEL produces a single macropulse of RF radiation rather than random, par tially overlapping, macropulses . A scheme of the mode-locked FEL oscillator i shown in fig . 1 . The accelerating potential is suppliei by a Marx generator (Physics International Pulsera 615 MR) . The continuous electron beam is generate by a thermionically emitting, electrostatically focuse( Pierce-type electron gun (250 kV, 250 A) from a SLAB it- 3.2 m (wovegu i de or coox ) --~ To osc+lloscol
WMAVA
PIN
WIGGLER
solenoid To MARX power suppy
w~~woo~~~~
0
27 MHz modulatic Isolotor
WVAVd 'VAVAVAVAVAVAV/ ThermiornC Pierce gun -2.25m-
Fig . l . The mode-locked FEL oscillator experimental setup
1992 - Elsevier Science Publishers B.V. All rights reserved
E. Jerby et al. / Mode-locked FEL oscillator
klystron (Model 343). An emittance selector is used to limit the beam current to - 1 A. The actual electron beam energy in this experiment is - 150 kcV, and the voltage droop is - 1 kV/p.s. An assembly of focusing coils transports the electron beam into the rectangular stainless steel drift tube (0.40 in. x 0.90 in.), which also acts as the wavcguidc for the electromagnetic radiation . The beam is contained by a uniform 1.6 kG axial magnetic field produced by a solenoid. The 65-period circularly-polarized magnetic wiggler is generated by bifilar conductors [20]. It has a period of lw = 3.5 cm and an amplitude of BW = 200-400 G. Since an aperture limits the size of the electron beam to r,, = 0.071,,., the wiggler field appears nearly sinusoidal to the drifting electrons. At the wiggler entrance a slowly increasing field amplitude is produced by resistively loading the first six periods of the wiggler magnet . The 2.7-m-long drift tube acts as a rectangular wavcguidc whose fundamental TIE,, mode has a cutoff frequency of 6.6 GHz . The system is operated in a frequency range between 8 and 11 GHz. At those frequencies the empty wavcguidc can support only the fundamental (TE ,) mode, all higher modes being evanescent . The output of the FEL is injected back to its input by a feedback line which forms a ring cavity loop. The perimeter of the ring cavity is 7.6 m. One section of the ring (3.2 m) is the feedback line, which can be either a nondispersive coaxial line (RG-214), or a waveguide section (WR-90). The modulation of the feedback attenuation is done by a PIN diode modulator (M.A. 8319-1 X23). The loss of the ring cavity (without the modulation) is 10 dB for a coaxial line, or 5.5 dB for a waveguide feedback section . The single-pass FEL gain varies so that the overall system net-gain is less than 3 dB. It is in this low net-gain operating regime that all of our measurements are carried out, and where the periodic RF macropulses are the clearest. In order to observe them, the radiation field of the ring cavity is sampled by means of a 17.2 dB directional-coupler and then measured with a crystal detector (HP423A) calibrated to 0.12 mW/mV at 100 mV. The signals are recorded by a digital oscilloscope (HP 54516A). This experimental setup resembles that of our previous experiment [18,19] . The main changes in the cavity are the installation of a PIN diode variable attenuator (modulated in 27 MHz), the removal of the RF filters, and the optional operation with a coaxial line in the feedback section . Another major modification in the present experiment is the reduction of the Marx voltage droop from 5 kV/Rs to 1 kV/Ws. The mode-locked FEL oscillator emits a clear single macropulse of RF radiation in each shot . The observed signals resemble the FEL irnpi.lse response computed
115
0.08
u
0.06
a ô 0.04 0
û
0.02
0.00 li 200
300
400
500
600
700
800
900
Fig . 2. A typical macropulse output signal of a mode-locked FEL with a coaxial line ring cavity. in ref. [ 19]. Fig . 2 shows an example of the FEL output with a coaxial feedback line. A single clear macropulse is observed with a period of - 37 ns and a micropulse width of - 3.5 ns. Taking into account the detector nonlinear response, the width of the RF micropulse is estimated to be - 2.8 ns. Fig . 3 shows a typical macropulse with a waveguide section as a feedback loop. The micropulse width in this case is twice as wide as with the coaxial line, as expected due to the waveguide dispersion. In the experimental results shown in fig. 2, for a coaxial feedback, line. the micropulse width (- 2.8 ns) 0.07 0.06 ,-, 0.05
u
a 0.04 0 U
v
v0
0_03 0.02 0.01 0.00 500
V~UÛÛÜÜ 1500
1000 t
2000
[r,S ]
Fig. 3 . A typical output of a mode-locked FEL with a waveguide ring cavity. 111. FEL EXPERIMENTS
E. Jerhy et al. / Moule-lockeol FEL oscillator
is close to the slippage time ( - 2.5 ns) computed in ref. [19]. The FEL response to a current impulse (namely, its temporal Greens function) is a finite pulse of radiation at the FEL resonance frequency. Its pulse width is
where t°- is the axial electron beam velocity, rg is the radiation group velocity, and L , is the wiggler length . The pulse width r,, known as the FEL slippage time (1), is the difference between the propagation times of an electron and a wave packet traveling along the wiggler axis. In the FEL oscillator, this pulse is circulating in the cavity and reamplified in each round-trip. As shown in ref. [191, this pulse tends to preserve its width, and its widening due to the waveguide dispersion is balanced by the FEL gain and phase shift. The
mode-locking
mechanism
enables
the
mi-
within short time-n-iaulows synchronized with the micropuls,: round-trip frequency. This time dependence of the loop net-gain results in a single clear macropulse, unlike our previous experiments [18.191 . in which we observed partially overlapping macropulses. In the frequency domain . the longitudinal modes under the FEL gain curve (whose cropulse train to evolve only
bandwidth is - 1/-.,) are locked : thus. their composition in the time domain results in narrow pulses with a macropulse width limited by the slippage time .
The clarity of the experimental results and the
theoretical analysis leads us to conclude that the effect observed has a general validity, and that it can be applicable: to other FEL oscillators, and in particular those with continuous electron beams . The electrostatic accelerator FEL [161 . for instance, has a special importance because of its unique CW operating mode and its potential spectral purity . Implementation of a mode-locking mechanism in such an FEL may extend its capabilities with a feature of macropulse operation, in addition to its CW mode . The width of the modelocked micropulses can reach. i n principle, the slippage time (1) which is given in the relativistic limit by ®, = L,,l2y_ c . where c is the speed of light, and y- _ [I - (t- ./C)']-' ' 2 is an axial relativistic factor . With the parameters of a typical electrostatic accelerator FEL (L,, = 3 m, y_ = 10) the expected macropulse width
is 50 ps. Much shorter pulses can be obtained with shorter wigglers and/or higher electron energies. In conclusion, the mode-locked FEL oscillator demonstrated in this experiment reveals aspects of the FEL Physics in the time domain . In practice, it may lead to applications at shorter wavelength, such as a two-mode (CW/short-pulse) FEL .
References [1] C .C. Cutler, Proc. IRE 43 (1955) 140 . [?] L.A. Glasser and H .A . Haus, IEEE Trans . Microwave Theory and Tech . 26 (1978) 62 . [31 P.W . Smith, Proc. IEEE 58 (1970) 1342, for recent studies on ultrafast laser phenomena . see the special issue of IEEE J . Quantum Electron . 25 (1989), [4] P .W. Smith . IEEE J . Quantum Electron . 3 (1967) 627. [5j J .R. Fontana. IEEE J. Quantum Electron . 8 (1972) 699 . [6] D.J . Kuizenga and A.E . Siegman, IEEE J . Quantum Electron . 6 (1970) 694. [7] S. Basu and R .L . Byer. IEEE J . Quantum Electron. 26 (1990) 149. [8] L .F. Mollenauer and R .H . Stolen . Opt. Let( . 9 (1984) 13 . [9] R .W. Warren. B.E . Newna m and J .C . Goldstein, IEEE J . Quantum Electron . 21 (1985) 882 . [10] J .C. Goldstein, B .W. Newnam . R.W. Warre n and R.L. Sheffield, Nucl . Instr . and Meth . A250 (1986) 4 . [11] J . Masud. T .C. Marshall, S .P. Schlesinger and F .G . Yee . Phys . Rev. Lett . 5 6 (1986) 1567 . [12] J .W. Dodd and T .C . Marshall, Nucl . Instr. and Meth . A296 (1990) 4. [ 13] B .A . Richman . J .M .J . Made y and E . Szarmes, Phys . Rev . Lett . 63 (1989) 1682 . 1141 T . Kawamura . K. Toyoda and M . Kawai . Appl . Phys . Lett . 5 1 (1987) 795 . [151 Y . Kawamura . B .C. Lee. M . Kawa i and K . Toyoda, Phys . Rev . Lett . 67 (1991) 832 . [16] A . Amir, R .J . Ho, F . Kielmann, J . Mertz and L .R. Elias, Nucl . Instr. and Meth . A272 (1988) 174 . [17] S .V . Benson, J .M .J . Madey, E .B . Szarmes, A . Bhowmik, P . Metty and M . Curtin, Nucl . Instr. and Meth . A296 (19911) 762 . [181 E . Jerby, G . Bekefi and J . Wurtele, Phvs . Rev . Lett . 66 (1991) 2068, also Nucl . Instr. and Meth . A304 (1991) 1(17 . [191 E . Jerby, G . Bekefi and J . Wurtele, IEEE J . Quantum Electron . 27 (1991) 2512 . [20] J . Fajans, G . Bekefi, Y .Z . Yin and B. Lax, Phys . Fluids 28 (1985) 1995 .