- Email: [email protected]
A comparison between the fundamental and third harmonic interactions in X-ray free-electron lasers
Nuclear Instruments and Methods in Physics Research A 475 (2001) 377–380
A comparison between the fundamental and third harmonic interactions in X-ray free-electron lasers H.P. Freund* Science Applications International Corp. 1710 Goodridge Drive, McLean, VA 22102, USA
Abstract ( X-rays. However, the interaction is Free-electron lasers are under study as 4th generation light sources for 1.5 A extremely sensitive to the beam quality, and it is useful to consider alternate schemes. The scheme discussed here is to ( X-rays using the fundamental interaction and rely on nonlinear 3rd harmonic generation to reach 1.5 A. ( generate 4.5 A The specific example discussed corresponds to the Linac Coherent Light Source proposed at the Stanford Linear Accelerator Center. r 2001 Elsevier Science B.V. All rights reserved. PACS: 41.60.Cr; 52.75.Ms Keywords: Free-electron lasers; Harmonics; Nonlinear harmonics
Studies are in progress for a free-electron laser (FEL) to serve as a 4th generation light source. The design goal of the Linac Coherent Light ( X-rays using Source (LCLS) is to generate 1.5 A the fundamental interaction [1]. However, the interaction is extremely sensitive to the beam quality; hence, it is useful to consider alternate schemes. One alternate scheme considered here is ( X-rays using the fundamental to generate 4.5 A and to rely on nonlinear 3rd harmonic generation ( At this longer wavelength, the [2,3] to reach 1.5 A. fundamental and 3rd harmonic are less sensitive to beam quality. These two schemes are compared for parameters similar to the nominal design parameters of the LCLS. The analysis is restricted to the case where *Tel.: +1-202-767-0034; fax: +1-202-734-1280. E-mail address: [email protected] (H.P. Freund).
( fundamental interaction is achieved by the 4.5 A increasing the wiggler period while holding other parameters fixed. We then study the sensitivity of both configurations to increases in the energy spread of the beam. The advantage of using a longer wiggler period is that both the wiggler Kvalue and the gap can be increased simultaneously. The increase in K is responsible for the reduced sensitivity to beam quality while the increase in the gap permits the use of a larger drift tube which reduces the influence of wakefields on the beam. However, alternate schemes such as reducing the beam energy are also possible. The MEDUSA simulation code [2,4] is used to study both schemes. MEDUSA is a three-dimensional, multi-frequency simulation code where the electromagnetic field is represented as a superposition of Gauss-Hermite modes. The electron beam is modeled as a Gaussian with a matched
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 5 4 3 - 1
H.P. Freund / Nuclear Instruments and Methods in Physics Research A 475 (2001) 377–380
beam radius. The field equations are integrated simultaneously with the 3D Lorentz force equations. A comparison between MEDUSA and four other FEL simulation codes, as well a linear analytic theory, has shown good agreement [5]. The nonlinear harmonic generation mechanism is well-known in traveling-wave tubes and has been discussed in FELs [2,3,6–8]. As the fundamental grows, the beam develops microbunches that give rise to enhanced nonlinear harmonic growth and can become the dominant harmonic growth mechanism where the gain length scales inversely with the harmonic number. In addition, the harmonic powers generated by this process can be quite high and can reach 10% of the power in the fundamental. The essential point is that the harmonic growth and saturation are controlled by the fundamental. This contrasts with the linear harmonic instability that is more sensitive to beam and wiggler quality than the fundamental. Hence, any decreased sensitivity of the fundamental to beam and wiggler quality also results in a decreased sensitivity of the nonlinear harmonic mechanism. A paper investigating the sensitivity of the nonlinear harmonic generation to wiggler imperfections is in preparation. The issue considered here is the sensitivity to beam energy spread. The nominal LCLS design employs a 14.35 GeV/ 3400 A electron beam with a normalized emittance of 1.5p mm-mrad and slice and global energy spreads of 0.006% and 0.02%, respectively. The wiggler has an amplitude of 13.2 kG and a period of 3.0 cm. We achieve operation on the funda( by the simple expedient of mental at 4.5 A increasing the wiggler period from 3.0 to 4.44 cm. Since the amplitude is fixed, this yields an increased K=5.47. The electron beam is unchanged, although variations in the energy spread are considered for both cases. No further optimi( design is attempted. In zation of the 4.5 A particular, the wiggler amplitude can be increased by using longer wiggler periods, and this results in further increases in K; however, at the cost of increases in the beam energy. One difference in this analysis from the LCLS design is that a single long parabolic-pole face (PPF) wiggler is considered for simplicity, since no
additional focusing is required. However, the betatron period for the PPF wiggler is of the order of 72 m. The actual wiggler design makes use of multiple wigglers with strong focusing resulting in (1) a shorter betatron period, (2) shorter gain length and (3) a reduced sensitivity to beam quality relative to the single PPF wiggler model. The PPF model, however, provides a qualitative picture of the nonlinear harmonic mechanism. First, consider a comparison of the performance ( designs for the ideal limit of of the 1.5 and 4.5 A zero beam energy spread. While the LCLS is to be used in Self-Amplified Spontaneous Emission (SASE) mode, it is convenient to consider an amplifier model where a drive power of 30 kW is ( for the two cases. assumed at either 1.5 or 4.5 A This permits the determination of the gain lengths and saturated powers to be expected in the SASE design. The results of the simulation are shown in Fig. 1. Note that although harmonics are also ( fundamental case, these associated with the 1.5 A are not included in this case since our primary ( interest is the performance at 1.5 A. ( The saturated power for the nominal 1.5 A design is about 12.8 GW with a gain length of 6.21 m. This is in reasonable agreement with a linear theory [9], which predicts a gain length of 5.40 m and a saturated power of 17.4 GW. The ( is 26.8 GW, which is higher peak power at 4.5 A ( design case due to the increased K than the 1.5 A
10
10
4.5 Å
8
10
Power (W)
378
1.5 Å harmonic
6
10
1.5 Å fundamental
4
10
2
10
∆E /E = 0% b
0
10
0
20
40
60
80
100
b
120
140
z (m) ( designs Fig. 1. Growth of the fundamental for the 1.5 and 4.5 A ( and of the 3rd harmonic of the 4.5 A interaction for zero energy spread.
379
H.P. Freund / Nuclear Instruments and Methods in Physics Research A 475 (2001) 377–380
10
10
4.5 Å 8
Power (W)
10
1.5 Å harmonic
6
10
1.5 Å fundamental
4
10
2
10
∆E /E = 0.02% b
0
10
0
20
40
60
80
b
100
120
140
z (m) ( designs Fig. 3. Growth of the fundamental for the 1.5 and 4.5 A ( interaction for a 0.02% and of the 3rd harmonic of the 4.5 A energy spread.
10
10
4.5 Å
8
10
Power (W)
value, and the gain length of 5.09 m is shorter. The gain length for the 3rd harmonic is about 1.67 m, which is approximately one-third that at the fundamental, as expected [2]. The harmonic power is a somewhat more ambiguous quantity. Typically, the nonlinear bunching at the harmonic causes the harmonic power to peak at a point just prior to the saturation of the fundamental. Subsequently, the harmonic power decreases due to overbunching but typically recovers and reaches its peak value somewhere downstream from the peak power point of the fundamental. For this example, the harmonic power peaks after a distance of 92 m at a power of 737 MW (about 2.8% of the fundamental power). However, the asymptotic value for the harmonic power can exceed this figure. Because of this ambiguity in the optimal harmonic power point, it is more useful to illustrate the effects of increasing energy spread. To this end, the corresponding evolution of the ( fundamental and the 4.5 A ( powers in the 1.5 A ( fundamental and 1.5 A harmonic for energy spreads of 0.01%, 0.02%, and 0.03% are shown in Figs. 2–4, respectively. First note that the gain ( lengths (saturation power levels) for the 1.5 A fundamental are 6.64 m (12.2 GW), 8.04 m (8.35 GW), and 10.07 m (5.70 GW) for DE b =E b =0.01%, 0.02%, and 0.03%, respectively. These are longer (lower) than the gain lengths
1.5 Å harmonic
6
10
1.5 Å fundamental
4
10
2
10
∆E /E = 0.03% b
0
10
0
30
60
90
120
b
150
180
z (m) ( designs Fig. 4. Growth of the fundamental for the 1.5 and 4.5 A ( interaction for a 0.03% and of the 3rd harmonic of the 4.5 A energy spread.
10
Power (W)
10
10
8
10
6
10
4
10
2
10
0
4.5 Å
1.5 Å harmonic 1.5 Å fundamental
∆E /E = 0.01% b
0
20
40
60
80
100
b
120
140
z (m) ( designs Fig. 2. Growth of the fundamental for the 1.5 and 4.5 A ( and of the 3rd harmonic of the 4.5 A interaction for a 0.01% energy spread.
( fundamental which are (power levels) for the 4.5 A 5.13 m (22.8 GW), 5.87 m (21.2 GW), and 7.15 m (17.5 GW) for DE b =E b =0.01%, 0.02%, and ( harmonic is 0.03%. The gain for the 1.5 A ( fundaapproximately one-third that of the 4.5 A mental and are 1.71, 1.94 and 2.17 m for the three choices of energy spread. In terms of the output power, it is clear that the nominal LCLS design ( harmonic for energy exceeds that of the 1.5 A spreads up to 0.03%. However, the saturation ( lengths for the nominal design for the 1.5 A fundamental exceed those for the harmonic design,
380
H.P. Freund / Nuclear Instruments and Methods in Physics Research A 475 (2001) 377–380
( harmonic are and the power levels for the 1.5 A appreciable and may become competitive if the achievable energy spread exceeds 0.03%. In summary, this study indicates that the ( for the nominal LCLS expected powers at 1.5 A design exceed those to be expected from harmonics for energy spreads below about 0.03%. This is for the case in which the beam energy and wiggler amplitude are held fixed while the wiggler period is increased; however, this conclusion may change if the harmonic is generated using lower beam energies or higher wiggler amplitudes. While the present analysis is based upon the natural focusing provided by a PPF wiggler, the basic physics applies to a strong focusing system as well. Computational work was supported by the Advanced Technology Group at Science Applications International Corporations under IR&D subproject 01-0060-73-0890-000.
References [1] NTIS Doc. No. DE98059292 (LCLS Design Study Group, ‘‘LCLS Design Report,’’ April 1998), National Technical Information Services, US Department of Commerce, 5285 Port Royal Road, Springfield, VA 22161. [2] H.P. Freund, et al., IEEE J. Quantum Electron. 36 (2000) 275. [3] Z.H. Huang, K.J. Kim, Nucl. Instr. and Meth. A 475 (2001) 112, this issue. [4] H.P. Freund, T.M. Antonsen Jr., Principles of Freeelectron Lasers, 2nd Edition, Chapman & Hall, London, 1996. [5] S.G. Biedron, et al., Nucl. Instr. and Meth. A 445 (2000) 110. [6] R. Bonifacio, et al., Nucl. Instr. and Meth. A 293 (1990) 627. [7] R. Bonifacio, et al., Nucl. Instr. and Meth. A 296 (1990) 787. [8] R. Bonifacio, R. Corsini, P. Pierini, Phys. Rev. A 45 (1992) 4091. [9] M. Xie, Proceedings of the IEEE 1995 Particle Accelerator Conference, IEEE Cat. No. 95CH35843, 1995, p. 183.
A comparison between the fundamental and third harmonic interactions in X-ray free-electron lasers H.P. Freund* Science Applications International Corp. 1710 Goodridge Drive, McLean, VA 22102, USA
Abstract ( X-rays. However, the interaction is Free-electron lasers are under study as 4th generation light sources for 1.5 A extremely sensitive to the beam quality, and it is useful to consider alternate schemes. The scheme discussed here is to ( X-rays using the fundamental interaction and rely on nonlinear 3rd harmonic generation to reach 1.5 A. ( generate 4.5 A The specific example discussed corresponds to the Linac Coherent Light Source proposed at the Stanford Linear Accelerator Center. r 2001 Elsevier Science B.V. All rights reserved. PACS: 41.60.Cr; 52.75.Ms Keywords: Free-electron lasers; Harmonics; Nonlinear harmonics
Studies are in progress for a free-electron laser (FEL) to serve as a 4th generation light source. The design goal of the Linac Coherent Light ( X-rays using Source (LCLS) is to generate 1.5 A the fundamental interaction [1]. However, the interaction is extremely sensitive to the beam quality; hence, it is useful to consider alternate schemes. One alternate scheme considered here is ( X-rays using the fundamental to generate 4.5 A and to rely on nonlinear 3rd harmonic generation ( At this longer wavelength, the [2,3] to reach 1.5 A. fundamental and 3rd harmonic are less sensitive to beam quality. These two schemes are compared for parameters similar to the nominal design parameters of the LCLS. The analysis is restricted to the case where *Tel.: +1-202-767-0034; fax: +1-202-734-1280. E-mail address: [email protected] (H.P. Freund).
( fundamental interaction is achieved by the 4.5 A increasing the wiggler period while holding other parameters fixed. We then study the sensitivity of both configurations to increases in the energy spread of the beam. The advantage of using a longer wiggler period is that both the wiggler Kvalue and the gap can be increased simultaneously. The increase in K is responsible for the reduced sensitivity to beam quality while the increase in the gap permits the use of a larger drift tube which reduces the influence of wakefields on the beam. However, alternate schemes such as reducing the beam energy are also possible. The MEDUSA simulation code [2,4] is used to study both schemes. MEDUSA is a three-dimensional, multi-frequency simulation code where the electromagnetic field is represented as a superposition of Gauss-Hermite modes. The electron beam is modeled as a Gaussian with a matched
0168-9002/01/$ - see front matter r 2001 Elsevier Science B.V. All rights reserved. PII: S 0 1 6 8 - 9 0 0 2 ( 0 1 ) 0 1 5 4 3 - 1
H.P. Freund / Nuclear Instruments and Methods in Physics Research A 475 (2001) 377–380
beam radius. The field equations are integrated simultaneously with the 3D Lorentz force equations. A comparison between MEDUSA and four other FEL simulation codes, as well a linear analytic theory, has shown good agreement [5]. The nonlinear harmonic generation mechanism is well-known in traveling-wave tubes and has been discussed in FELs [2,3,6–8]. As the fundamental grows, the beam develops microbunches that give rise to enhanced nonlinear harmonic growth and can become the dominant harmonic growth mechanism where the gain length scales inversely with the harmonic number. In addition, the harmonic powers generated by this process can be quite high and can reach 10% of the power in the fundamental. The essential point is that the harmonic growth and saturation are controlled by the fundamental. This contrasts with the linear harmonic instability that is more sensitive to beam and wiggler quality than the fundamental. Hence, any decreased sensitivity of the fundamental to beam and wiggler quality also results in a decreased sensitivity of the nonlinear harmonic mechanism. A paper investigating the sensitivity of the nonlinear harmonic generation to wiggler imperfections is in preparation. The issue considered here is the sensitivity to beam energy spread. The nominal LCLS design employs a 14.35 GeV/ 3400 A electron beam with a normalized emittance of 1.5p mm-mrad and slice and global energy spreads of 0.006% and 0.02%, respectively. The wiggler has an amplitude of 13.2 kG and a period of 3.0 cm. We achieve operation on the funda( by the simple expedient of mental at 4.5 A increasing the wiggler period from 3.0 to 4.44 cm. Since the amplitude is fixed, this yields an increased K=5.47. The electron beam is unchanged, although variations in the energy spread are considered for both cases. No further optimi( design is attempted. In zation of the 4.5 A particular, the wiggler amplitude can be increased by using longer wiggler periods, and this results in further increases in K; however, at the cost of increases in the beam energy. One difference in this analysis from the LCLS design is that a single long parabolic-pole face (PPF) wiggler is considered for simplicity, since no
additional focusing is required. However, the betatron period for the PPF wiggler is of the order of 72 m. The actual wiggler design makes use of multiple wigglers with strong focusing resulting in (1) a shorter betatron period, (2) shorter gain length and (3) a reduced sensitivity to beam quality relative to the single PPF wiggler model. The PPF model, however, provides a qualitative picture of the nonlinear harmonic mechanism. First, consider a comparison of the performance ( designs for the ideal limit of of the 1.5 and 4.5 A zero beam energy spread. While the LCLS is to be used in Self-Amplified Spontaneous Emission (SASE) mode, it is convenient to consider an amplifier model where a drive power of 30 kW is ( for the two cases. assumed at either 1.5 or 4.5 A This permits the determination of the gain lengths and saturated powers to be expected in the SASE design. The results of the simulation are shown in Fig. 1. Note that although harmonics are also ( fundamental case, these associated with the 1.5 A are not included in this case since our primary ( interest is the performance at 1.5 A. ( The saturated power for the nominal 1.5 A design is about 12.8 GW with a gain length of 6.21 m. This is in reasonable agreement with a linear theory [9], which predicts a gain length of 5.40 m and a saturated power of 17.4 GW. The ( is 26.8 GW, which is higher peak power at 4.5 A ( design case due to the increased K than the 1.5 A
10
10
4.5 Å
8
10
Power (W)
378
1.5 Å harmonic
6
10
1.5 Å fundamental
4
10
2
10
∆E /E = 0% b
0
10
0
20
40
60
80
100
b
120
140
z (m) ( designs Fig. 1. Growth of the fundamental for the 1.5 and 4.5 A ( and of the 3rd harmonic of the 4.5 A interaction for zero energy spread.
379
H.P. Freund / Nuclear Instruments and Methods in Physics Research A 475 (2001) 377–380
10
10
4.5 Å 8
Power (W)
10
1.5 Å harmonic
6
10
1.5 Å fundamental
4
10
2
10
∆E /E = 0.02% b
0
10
0
20
40
60
80
b
100
120
140
z (m) ( designs Fig. 3. Growth of the fundamental for the 1.5 and 4.5 A ( interaction for a 0.02% and of the 3rd harmonic of the 4.5 A energy spread.
10
10
4.5 Å
8
10
Power (W)
value, and the gain length of 5.09 m is shorter. The gain length for the 3rd harmonic is about 1.67 m, which is approximately one-third that at the fundamental, as expected [2]. The harmonic power is a somewhat more ambiguous quantity. Typically, the nonlinear bunching at the harmonic causes the harmonic power to peak at a point just prior to the saturation of the fundamental. Subsequently, the harmonic power decreases due to overbunching but typically recovers and reaches its peak value somewhere downstream from the peak power point of the fundamental. For this example, the harmonic power peaks after a distance of 92 m at a power of 737 MW (about 2.8% of the fundamental power). However, the asymptotic value for the harmonic power can exceed this figure. Because of this ambiguity in the optimal harmonic power point, it is more useful to illustrate the effects of increasing energy spread. To this end, the corresponding evolution of the ( fundamental and the 4.5 A ( powers in the 1.5 A ( fundamental and 1.5 A harmonic for energy spreads of 0.01%, 0.02%, and 0.03% are shown in Figs. 2–4, respectively. First note that the gain ( lengths (saturation power levels) for the 1.5 A fundamental are 6.64 m (12.2 GW), 8.04 m (8.35 GW), and 10.07 m (5.70 GW) for DE b =E b =0.01%, 0.02%, and 0.03%, respectively. These are longer (lower) than the gain lengths
1.5 Å harmonic
6
10
1.5 Å fundamental
4
10
2
10
∆E /E = 0.03% b
0
10
0
30
60
90
120
b
150
180
z (m) ( designs Fig. 4. Growth of the fundamental for the 1.5 and 4.5 A ( interaction for a 0.03% and of the 3rd harmonic of the 4.5 A energy spread.
10
Power (W)
10
10
8
10
6
10
4
10
2
10
0
4.5 Å
1.5 Å harmonic 1.5 Å fundamental
∆E /E = 0.01% b
0
20
40
60
80
100
b
120
140
z (m) ( designs Fig. 2. Growth of the fundamental for the 1.5 and 4.5 A ( and of the 3rd harmonic of the 4.5 A interaction for a 0.01% energy spread.
( fundamental which are (power levels) for the 4.5 A 5.13 m (22.8 GW), 5.87 m (21.2 GW), and 7.15 m (17.5 GW) for DE b =E b =0.01%, 0.02%, and ( harmonic is 0.03%. The gain for the 1.5 A ( fundaapproximately one-third that of the 4.5 A mental and are 1.71, 1.94 and 2.17 m for the three choices of energy spread. In terms of the output power, it is clear that the nominal LCLS design ( harmonic for energy exceeds that of the 1.5 A spreads up to 0.03%. However, the saturation ( lengths for the nominal design for the 1.5 A fundamental exceed those for the harmonic design,
380
H.P. Freund / Nuclear Instruments and Methods in Physics Research A 475 (2001) 377–380
( harmonic are and the power levels for the 1.5 A appreciable and may become competitive if the achievable energy spread exceeds 0.03%. In summary, this study indicates that the ( for the nominal LCLS expected powers at 1.5 A design exceed those to be expected from harmonics for energy spreads below about 0.03%. This is for the case in which the beam energy and wiggler amplitude are held fixed while the wiggler period is increased; however, this conclusion may change if the harmonic is generated using lower beam energies or higher wiggler amplitudes. While the present analysis is based upon the natural focusing provided by a PPF wiggler, the basic physics applies to a strong focusing system as well. Computational work was supported by the Advanced Technology Group at Science Applications International Corporations under IR&D subproject 01-0060-73-0890-000.
References [1] NTIS Doc. No. DE98059292 (LCLS Design Study Group, ‘‘LCLS Design Report,’’ April 1998), National Technical Information Services, US Department of Commerce, 5285 Port Royal Road, Springfield, VA 22161. [2] H.P. Freund, et al., IEEE J. Quantum Electron. 36 (2000) 275. [3] Z.H. Huang, K.J. Kim, Nucl. Instr. and Meth. A 475 (2001) 112, this issue. [4] H.P. Freund, T.M. Antonsen Jr., Principles of Freeelectron Lasers, 2nd Edition, Chapman & Hall, London, 1996. [5] S.G. Biedron, et al., Nucl. Instr. and Meth. A 445 (2000) 110. [6] R. Bonifacio, et al., Nucl. Instr. and Meth. A 293 (1990) 627. [7] R. Bonifacio, et al., Nucl. Instr. and Meth. A 296 (1990) 787. [8] R. Bonifacio, R. Corsini, P. Pierini, Phys. Rev. A 45 (1992) 4091. [9] M. Xie, Proceedings of the IEEE 1995 Particle Accelerator Conference, IEEE Cat. No. 95CH35843, 1995, p. 183.