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Multi-mode interactions in an FEL oscillator
Nuclear Instruments and Methods in Physics Research A 445 (2000) 101}104
Multi-mode interactions in an FEL oscillator Zhi-Wei Dong*, Jiro Kitagaki, Kai Masuda, Tetsuo Yamazaki, Kiyoshi Yoshikawa Institute of Advanced Energy, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan
Abstract A 3D time-dependent FEL oscillator simulation code has been developed by using the transverse mode spectral method to analyze interaction among transverse modes. The competition among them in an FEL oscillator was investigated based on the parameters of LANL FEL experiments. It is found that under typical FEL oscillator operation conditions, the TEM mode is dominant, and the e!ects of other transverse modes can be negligible. 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 41.60.C Keywords: 3D space}time interaction; Mode competition; Transverse modes
1. Introduction Time dependent or slippage e!ects are very important in an RF LINAC-based FEL, since they play a very important role in the evolution of sideband, lethargy, superradiance and so on. Usually, these longitudinal-dominated e!ects were so far treated by the 1D time-dependent FEL code [1]. However, the actual FEL interaction is space} time dependent 3D processes including transverse mode competition, and thus a 3D time-dependent FEL code is essential for both a full understanding of FEL physics and facility designing. Since a 3D time-dependent simulation would be very time consuming if the wave equation were solved straightforward either in the time domain [2] or in frequency domain by FFT [3], a 3D time-dependent
FEL code is developed by applying the transverse mode spectral method [4] in order to make a simulation by a PC within a tolerable CPU time. In this paper, we examine the e!ects of the competition among the transverse modes on the FEL processes.
2. Computational model In the development of the present 3D code with the transverse mode spectral method [4], the following equations are adopted. 2.1. Wave equation The normalized radiation "eld can be expressed as
* Corresponding author. Tel.: #81-774-38-3443; fax: #81774-38-3449. E-mail address: [email protected] (Z.-W. Dong).
+ , a (x, y, z, t)" a (z, t) ) e( (x, y, z) 0 KL KL K L ;exp[i (k z!u t)]#c.c.
0168-9002/00/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 1 2 1 - 2
(1)
SECTION I.
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where a (z, t) is the normalized complex ampliKL tude of the transverse mode TEM , of the pattern KL e( with wave number k and frequency u, the KL x component of e( , G "g ) exp[!ih ]. The KL KL KL KL wave equation can be expressed as
R R F(k )a (x , y , z)GH e\ RH H H H KL # a "iC Rz cRt KL c H
(2)
where C"1/2 k (u/c)p , u is the electron plasma frequency, p is the e!ective area of the electron beam, and F(k )"J (k )!J (k ) H H H is the longitudinal coupling factor with k "k a (r )/8k c, 1()2"1/N , () is the H H H H H ensemble average, respectively. Without losing #exibility, the transverse modes can be expressed by Gaussian}Hermite expansion g "1/((2Km!(2Ln!)(2/p/= KL ;H (x )H (y ) e\ V >W . K L
(3)
h "1/2(x #y )m!(m#n#1)tg\m and m" KL (z!z )/z , ="= [1#m] , with x "(2x/=, 0 y "(2y/=, z "1/2k = , and } z "1/2¸ , 0 where H is the Hermite polynomial of the mth K order, z is the axial location of the minimum waist, = is the spot size, and z is the Rayleigh length, 0 respectively.
where 1 2 sums over all the transverse modes TEM , while the transverse motion of the elecKL trons can be expressed by betatron equations.
3. Simulation results In this paper the periodic boundary was assumed since the electron bunch is much longer than the slippage distance, and the particle number for each `electron slicea was chosen to be 128. All the simulation were performed for the following two cases, i.e. (a) with only one fundamental transverse mode (TEM ), and (b) with the "rst 36 transverse modes (from TEM up to TEM ). The e!ects of trans verse modes on FEL processes are examined for two operation regions, i.e., in a moderately saturated region and a deeply spiking region. First, we made simulation using the LANL's experimental data [2] with a relatively large output power coupling fraction (Q"0.92) so that the system may just reach the saturation, i.e., the sideband is supposed not to appear. It is found from the gain and intra-cavity powers shown in Fig. 1, that: (1) In
2.2. Electron motion The longitudinal electron motions are expressed by the following equations
R 1 R 1 # c " F(k )a (r )k H H H Rz v Rt 2c H z1a (r , q) cos t 2 H H
R 1 R k # h "k ! [1#a (r ) H H Rz v Rt 2c H !F(k )a (r )1a (r , q) sin t 2 H H H H #1a(r, q)#1cb 2] H ,@H
(4)
Fig. 1. Gain (top) and intra-cavity power (bottom) versus passes for I"100 A and Q"0.92. Dotted line, fundamental transverse mode; solid line, multiple transverse modes.
Z.-W. Dong et al. / Nuclear Instruments and Methods in Physics Research A 445 (2000) 101}104
Fig. 2. Intra-cavity power with I"400 A and Q"0.98. Dotted line, fundamental transverse mode; solid line, multiple transverse modes.
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the small signal region the competition among the transverse modes makes the TEM gain smaller than the corresponding one in the fundamental mode operation. (2) After entering the exponential gain region, the gain of the fundamental mode gradually dominates over other higher transverse modes, and in both cases of above (a) and (b), the gains tend to approach the same value. (3) Output power mainly comes from the TEM mode, and the total contribution from other higher transverse modes is less than 2%. (4) The multiple transverse mode competition makes the optical build-up time here a little longer by 3}5 passes. It is also found by the simulation that the resonant frequency shifts from the higher transverse modes, TEM , are very KL
Fig. 3. Spectra for I"400 A, and Q"0.98. Left, fundamental transverse mode; right, multiple transverse modes.
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small, thus their e!ects on FEL spectra can be neglected. Consequently, in a moderately saturated FEL region, the competition among the transverse modes shows almost no e!ects on FEL interaction processes, except for a little longer optical build-up time. Next, we consider the e!ects of transverse modes when the system goes deeply into sideband and spiking regions. Fig. 2 shows the intra-cavity power for fundamental and multi-modes with beam currents equal to 400 A for Q"0.98 with the corresponding spectra in Fig. 3. It is seen that the power evolutions for the two cases are almost similar, and the corresponding spectra clearly show the chaotic spiking characteristics due to stronger sideband interaction. The normalized contribution from the largest 2 higher transverse modes, TEM and TEM , shown in Fig. 4, clearly shows less than 10% total contribution. From these results it is concluded that in a spiking FEL region the longitudinal competition or sideband instability prevails over the competition among transverse modes. Although the contribution from the higher transverse modes is found to gradually increase with the decrease of the power extraction fraction from the mirror and/or increase of the beam current, it is found to lie less than 10% of the total power.
4. Conclusions Under typical FEL oscillator operation conditions, the TEM mode is found dominant over
Fig. 4. Normalized power contribution from the largest two higher transverse modes for I"400 A and Q"0.98.
other transverse modes either in moderately saturated region or deeply spiking region. The effects of higher transverse modes can be found to be negligible, which indicates that it is not necessary to make a transverse mode selection.
References [1] J.C. Goldstein et al., SPIE Vol. 1045 (1989) 29. [2] B.D. McVey, Nucl. Instr. and Meth. A 250 (1986) 449. [3] T. Nakamura, Proceedings of the 14th LINAC Meeting in Japan, 1989, p. 236. [4] C.M. Tang, P.A. Sprangle, IEEE QE-21, No. 7 (1985) 970.
Multi-mode interactions in an FEL oscillator Zhi-Wei Dong*, Jiro Kitagaki, Kai Masuda, Tetsuo Yamazaki, Kiyoshi Yoshikawa Institute of Advanced Energy, Kyoto University, Gokasho, Uji, Kyoto 611-0011, Japan
Abstract A 3D time-dependent FEL oscillator simulation code has been developed by using the transverse mode spectral method to analyze interaction among transverse modes. The competition among them in an FEL oscillator was investigated based on the parameters of LANL FEL experiments. It is found that under typical FEL oscillator operation conditions, the TEM mode is dominant, and the e!ects of other transverse modes can be negligible. 2000 Published by Elsevier Science B.V. All rights reserved. PACS: 41.60.C Keywords: 3D space}time interaction; Mode competition; Transverse modes
1. Introduction Time dependent or slippage e!ects are very important in an RF LINAC-based FEL, since they play a very important role in the evolution of sideband, lethargy, superradiance and so on. Usually, these longitudinal-dominated e!ects were so far treated by the 1D time-dependent FEL code [1]. However, the actual FEL interaction is space} time dependent 3D processes including transverse mode competition, and thus a 3D time-dependent FEL code is essential for both a full understanding of FEL physics and facility designing. Since a 3D time-dependent simulation would be very time consuming if the wave equation were solved straightforward either in the time domain [2] or in frequency domain by FFT [3], a 3D time-dependent
FEL code is developed by applying the transverse mode spectral method [4] in order to make a simulation by a PC within a tolerable CPU time. In this paper, we examine the e!ects of the competition among the transverse modes on the FEL processes.
2. Computational model In the development of the present 3D code with the transverse mode spectral method [4], the following equations are adopted. 2.1. Wave equation The normalized radiation "eld can be expressed as
* Corresponding author. Tel.: #81-774-38-3443; fax: #81774-38-3449. E-mail address: [email protected] (Z.-W. Dong).
+ , a (x, y, z, t)" a (z, t) ) e( (x, y, z) 0 KL KL K L ;exp[i (k z!u t)]#c.c.
0168-9002/00/$ - see front matter 2000 Published by Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 0 ) 0 0 1 2 1 - 2
(1)
SECTION I.
102
Z.-W. Dong et al. / Nuclear Instruments and Methods in Physics Research A 445 (2000) 101}104
where a (z, t) is the normalized complex ampliKL tude of the transverse mode TEM , of the pattern KL e( with wave number k and frequency u, the KL x component of e( , G "g ) exp[!ih ]. The KL KL KL KL wave equation can be expressed as
R R F(k )a (x , y , z)GH e\ RH H H H KL # a "iC Rz cRt KL c H
(2)
where C"1/2 k (u/c)p , u is the electron plasma frequency, p is the e!ective area of the electron beam, and F(k )"J (k )!J (k ) H H H is the longitudinal coupling factor with k "k a (r )/8k c, 1()2"1/N , () is the H H H H H ensemble average, respectively. Without losing #exibility, the transverse modes can be expressed by Gaussian}Hermite expansion g "1/((2Km!(2Ln!)(2/p/= KL ;H (x )H (y ) e\ V >W . K L
(3)
h "1/2(x #y )m!(m#n#1)tg\m and m" KL (z!z )/z , ="= [1#m] , with x "(2x/=, 0 y "(2y/=, z "1/2k = , and } z "1/2¸ , 0 where H is the Hermite polynomial of the mth K order, z is the axial location of the minimum waist, = is the spot size, and z is the Rayleigh length, 0 respectively.
where 1 2 sums over all the transverse modes TEM , while the transverse motion of the elecKL trons can be expressed by betatron equations.
3. Simulation results In this paper the periodic boundary was assumed since the electron bunch is much longer than the slippage distance, and the particle number for each `electron slicea was chosen to be 128. All the simulation were performed for the following two cases, i.e. (a) with only one fundamental transverse mode (TEM ), and (b) with the "rst 36 transverse modes (from TEM up to TEM ). The e!ects of trans verse modes on FEL processes are examined for two operation regions, i.e., in a moderately saturated region and a deeply spiking region. First, we made simulation using the LANL's experimental data [2] with a relatively large output power coupling fraction (Q"0.92) so that the system may just reach the saturation, i.e., the sideband is supposed not to appear. It is found from the gain and intra-cavity powers shown in Fig. 1, that: (1) In
2.2. Electron motion The longitudinal electron motions are expressed by the following equations
R 1 R 1 # c " F(k )a (r )k H H H Rz v Rt 2c H z1a (r , q) cos t 2 H H
R 1 R k # h "k ! [1#a (r ) H H Rz v Rt 2c H !F(k )a (r )1a (r , q) sin t 2 H H H H #1a(r, q)#1cb 2] H ,@H
(4)
Fig. 1. Gain (top) and intra-cavity power (bottom) versus passes for I"100 A and Q"0.92. Dotted line, fundamental transverse mode; solid line, multiple transverse modes.
Z.-W. Dong et al. / Nuclear Instruments and Methods in Physics Research A 445 (2000) 101}104
Fig. 2. Intra-cavity power with I"400 A and Q"0.98. Dotted line, fundamental transverse mode; solid line, multiple transverse modes.
103
the small signal region the competition among the transverse modes makes the TEM gain smaller than the corresponding one in the fundamental mode operation. (2) After entering the exponential gain region, the gain of the fundamental mode gradually dominates over other higher transverse modes, and in both cases of above (a) and (b), the gains tend to approach the same value. (3) Output power mainly comes from the TEM mode, and the total contribution from other higher transverse modes is less than 2%. (4) The multiple transverse mode competition makes the optical build-up time here a little longer by 3}5 passes. It is also found by the simulation that the resonant frequency shifts from the higher transverse modes, TEM , are very KL
Fig. 3. Spectra for I"400 A, and Q"0.98. Left, fundamental transverse mode; right, multiple transverse modes.
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small, thus their e!ects on FEL spectra can be neglected. Consequently, in a moderately saturated FEL region, the competition among the transverse modes shows almost no e!ects on FEL interaction processes, except for a little longer optical build-up time. Next, we consider the e!ects of transverse modes when the system goes deeply into sideband and spiking regions. Fig. 2 shows the intra-cavity power for fundamental and multi-modes with beam currents equal to 400 A for Q"0.98 with the corresponding spectra in Fig. 3. It is seen that the power evolutions for the two cases are almost similar, and the corresponding spectra clearly show the chaotic spiking characteristics due to stronger sideband interaction. The normalized contribution from the largest 2 higher transverse modes, TEM and TEM , shown in Fig. 4, clearly shows less than 10% total contribution. From these results it is concluded that in a spiking FEL region the longitudinal competition or sideband instability prevails over the competition among transverse modes. Although the contribution from the higher transverse modes is found to gradually increase with the decrease of the power extraction fraction from the mirror and/or increase of the beam current, it is found to lie less than 10% of the total power.
4. Conclusions Under typical FEL oscillator operation conditions, the TEM mode is found dominant over
Fig. 4. Normalized power contribution from the largest two higher transverse modes for I"400 A and Q"0.98.
other transverse modes either in moderately saturated region or deeply spiking region. The effects of higher transverse modes can be found to be negligible, which indicates that it is not necessary to make a transverse mode selection.
References [1] J.C. Goldstein et al., SPIE Vol. 1045 (1989) 29. [2] B.D. McVey, Nucl. Instr. and Meth. A 250 (1986) 449. [3] T. Nakamura, Proceedings of the 14th LINAC Meeting in Japan, 1989, p. 236. [4] C.M. Tang, P.A. Sprangle, IEEE QE-21, No. 7 (1985) 970.