** Journal of Modern Physics** Vol.3 No.3(2012), Article ID:18205,4 pages DOI:10.4236/jmp.2012.33039

The Study and Investigation of Electron Trajectories in Free Electron Laser with Realizable Helical Wiggler and Ion Channel Guiding: A Computational Study

Institute for Advanced Studies, Tehran, Iran

Email: ^{*}mohammadalighorbani1983@yahoo.com

Received October 24, 2011; revised December 6, 2011; accepted December 29, 2011

**Keywords:** Free Electron Laser; Ion Channel; Realizable Helical Wiggler; Electron Trajectories; Velocity of Electron

ABSTRACT

In this paper, we analyze the electron trajectories in the free electron laser with the realizable helical wiggler and ion channel guiding, and achieve the stable and unstable states of electron orbit. The results indicate that the stability of electron trajectories (single electron) in the realizable helical wiggler investigated in the three-dimensional space differs from the ideal helical wiggler achieved previously in the two-dimensional coordinates. In the ideal state, the helical wiggler is considered to have only two components on the surface perpendicular to the electron beam’s trajectory, and the third component and latitudinal velocity of electron v_{x}_{,y} with respect to axial velocity v_{z} are neglected. This approximation finally leads to differences between theoretical and empirical results. Whereas regarding the realizable helical wiggler including three special components, and applying the latitudinal velocity of electron, this difference is expected to decline.

1. Introduction

In this paper, a detailed analysis of electron trajectories in a realizable helical wiggler in the free electron laser with ion-channel guiding is presented. We investigate the conditions for the stability of electron trajectories; and the calculations are illustrated. We indicate that there are differences between stable and unstable conditions for electron trajectories, and between the ideal 2D helical wiggler and 3D realizable helical wiggler. The ion channel with positive ions is analyzed as a guiding tool in order to improve the performance of free electron lasers with ideal helical wiggler [1-5] and ideal planar wiggler [6,7], the electron trajectories in ideal planar wiggler are computed [6,7]. When the electron beam transits this channel, the existence of positive ions create electrostatic field in the latitudinal direction, and cause the electron beam to focus in the direction of laser cylinder’s axis (active medium).

In this paper, the electron trajectories in the free electron laser with realizable helical wiggler and ion channel are investigated. The latitudinal electrostatic field due to positive ions with uniform density and static magnetic wiggler are considered. The relativistic electron-motion equation is solved. Finally, through employing the multinomial equations in the steady state, the stable and unstable electron trajectories including two orbital groups are obtained [8].

2. Results and Discussion

2.1. The Electron Trajectories

In this paper, for the first time, we proposed a novel analytical and numerical approach toward studying electron trajectories in free electron laser with realizable helical wiggler and ion channel guiding. In the free electron laser with the ideal helical magnetic wiggler, the magnetic wiggler has only two components in the directions of x and y, and the component z is neglected [8]. However, in the realizable wiggler, all the components are input to approach the laser’s real form.

The realizable helical magnetic wiggler in the cylindrical coordinates is defined as follows [8-23]:

(1)

where B_{W} is the wiggler amplitude, is the Bessel function, , , is the wiggler wave number, and.

The electrostatic field resulting from the ion channel can be written as [6,7]:

(2)

where is the positive ions’ density, and e is the absolute value of electrons’ charge. In the presence of these fields, the motion equation is as follows [8]:

(3)

where is the electron vector velocity, and c is the light velocity. Solving the motion equations in order to investigate electron trajectories in the Cartesian coordinates is difficult; accordingly, we employ the wiggler coordinates [9-23]. The realizable helical magnetic wiggler in this coordinate system is as the following form [8]:

(4)

Through inserting Equations (2) and (4) into Equation (3), the motion equation’s components are as follows:

(5)

(6)

(7)

(8)

(9)

where, , is the relativistic factor, m is the static electron’s mass. The solutions to Equations (5)-(9) in the steady state are as the forms below:

(10)

(11)

(12)

(13)

(14)

If or (latitudinal velocity) inclines toward zero, the steady-state results are fulfilled [1-5].

In regard to Equation (10), when the below condition exists:

(15)

The latitudinal velocity () dramatically increases. In other words, there is a resonance creating two orbital groups as the following forms:

The orbital group I:

The orbital group II:

2.2. The Stability of Trajectories

The stability of steady trajectories is achieved through inputting perturbations of, , , , , and around the steady state (and are constant) to Equations (5)-(8):

(16)

(17)

(18)

Through the derivation of Equations (16) and (17) with respect to time, and the replacement of Equation (18), the two equations are coupled together:

(19)

(20)

(21)

(22)

(23)

(24)

(25)

(26)

where and. Considering the oscillating commutation:

(27)

where is the commutation amplitude, and is its frequency. Finally, Equations (19) and (20) are transformed into the forms below:

(28)

(29)

The above equation is a second-order equation according to, and the stability condition is as the following form:

(30)

With reference to this condition (Equation (30)), Equation (27) is oscillating.

The graph of electron’s axial velocity (β_{II}) by for ideal wiggler is illustrated in Figure 1 [6,7]; Figure 2 indicates by for realizable wiggler (in both figures, we consider). In contrast with the ideal wiggler, there is instability in the orbital group II. Furthermore, the declined resonance state decreases the performance. It is anticipated that as the performance slumps on account of unstable trajectories in the realizable wiggler, the difference between theoretical and empirical results of ideal wiggler reduces [8].

3. Conclusion

In this paper, the steady electron trajectories in the free electron laser with the realizable helical magnetic wiggler and ion channel guiding are investigated using the

Figure 1. The diagram of β_{II} by for the ideal wiggler.

Figure 2. The diagram of β_{II} by for the realizable wiggler.

relativistic motion equation, and compared with the ideal state. In contrast with the ideal wiggler, in the realizable helical wiggler, the orbital group II has unstable trajectories. The unstable trajectories increase in proportion to the ideal wiggler, and the resonance region reduces. Therefore, it is expected that the difference between theoretical and empirical results of laser performance, observed in the ideal state, also decreases.

4. Acknowledgements

The work described in this paper was fully supported by grants from the Institute for Advanced Studies of Iran. The authors would like to express genuinely and sincerely thanks and appreciated and their gratitude to Institute for Advanced Studies of Iran.

REFERENCES

- P. Jha and P. Kumar, “Electron Trajectories and Gain in Free Electron Laser with Ion Channel Guiding,” IEEE Transactions on Plasma Science, Vol. 24, No. 6, 1996, pp. 1359-1363. doi:10.1109/27.553201
- P. Jha, P. Kumar, A. K. Upadhyaya and G. Raj, “Electric and Magnetic Wakefields in a Plasma Channel,” Physical Review Special Topics-Accelerators and Beams, Vol. 8, No. 8, 2005, pp. 071301-071307. doi:10.1103/PhysRevSTAB.8.071301
- P. Jha, P. Kumar and K. Pande, “Harmonic Generation in Free-Electron Laser with Circularly Polarized Wiggler and Ion-Channel Guiding,” IEEE Transactions on Plasma Science, Vol. 27, No. 2, 1999, pp. 637-642. doi:10.1109/27.772297
- P. Jha and P. Kumar, “Dispersion Relation and Growth in a Free-Electron Laser with Ion-Channel Guiding,” Physical Review E, Vol. 57, No. 2, 1998, pp. 2256-2261. doi:10.1109/27.772297
- P. Jha, P. Kumar, G. Raj and A. K. Upadhyaya, “Modulation Instability of Laser Pulse in Magnetized Plasma,” Physics of Plasmas, Vol. 12, No. 12, 2005, pp. 3104- 3109. doi:10.1063/1.2141399
- S. Lal, P. Kumar and P. Jha, “Free Electron Laser with Linearly Polarized Wiggler and Ion Channel Guiding,” Physics of Plasmas, Vol. 10, No. 7, 2003, pp. 3012-3016. doi:10.1063/1.1582475
- N. Wadhwani, P. Kumar and P. Jha, “Nonlinear Theory of Propagation of Intense Laser Pulses in Magnetized Plasma,” Physics of Plasmas, Vol. 9, No. 1, 2002, pp. 263-266. doi:10.1063/1.1421369
- H. P. Freund and T. M. Antonsen Jr., “Principles of FreeElectron Lasers,” Chapman & Hall, London, 1996.
- V. N. Litvinenko and Y. K. Wu, “Free Electron Lasers 2000,” Elsevier, Amsterdam, 2001.
- H. P. Freund and A. K. Ganguly, “Three-Dimensional Theory of the Free-Electron Laser in the Collective Regime,” Physical Review A, Vol. 28, No. 6, 1983, pp. 3438-3449. doi:10.1103/PhysRevA.28.3438
- A. K. Ganguly and H. P. Freund, “Nonlinear Analysis of Free-Electron-Laser Amplifiers in Three Dimensions,” Physical Review A, Vol. 32, No. 4, 1985, pp. 2275-2286. doi:10.1103/PhysRevA.32.2275
- H. P. Freund and A. K. Ganguly, “Nonlinear Analysis of Efficiency Enhancement in Free-Electron-Laser Amplifiers,” Physical Review A, Vol. 33, No. 2, 1986, pp. 1060- 1072. doi:10.1103/PhysRevA.33.1060
- H. P. Freund and A. K. Ganguly, “Effect of Beam Quality on the Free-Electron Laser,” Physical Review A, Vol. 34, No. 2, 1986, pp. 1242-1246. doi:10.1103/PhysRevA.34.1242
- A. K. Ganguly and H. P. Freund, “Three-Dimensional Simulation of the Raman Free-Electron Laser,” Physics of Fluids, Vol. 31, No. 2, 1988, pp. 387-393. doi:10.1063/1.866819
- H. P. Freund and A. K. Ganguly, “Nonlinear Simulation of a High-Power, Collective Free-Electron Laser,” IEEE Transactions on Plasma Science, Vol. 20, No. 3, 1992, pp. 245-255. doi:10.1109/27.142826
- A. K. Ganguly and H. P. Freund, “High-Efficiency Operation of Free-Electron-Laser Amplifiers,” IEEE Transactions on Plasma Science, Vol. 16, No. 2, 1988, pp. 167- 171. doi:10.1109/27.3810
- D. A. Kirkpatrick, G. Bekefi, A. C. Dirienzo, H. P. Freund and A. K. Ganguly, “A High Power, 600 μm Wavelength Free Electron Laser,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 285, No. 1-2, 1989, pp. 43-46. doi:10.1016/0168-9002(89)90422-1
- A. K. Ganguly, H. P. Freund and S. Ahn, “Nonlinear Theory of the Orbitron Maser in Three Dimensions,” Physical Review A, Vol. 36, No. 5, 1987, pp. 2199-2209. doi:10.1103/PhysRevA.36.2199
- P. M. Phillips, E. G. Zaidman, H. P. Freund, A. K. Ganguly and N. R. Vanderplaats, “Review of Two-Stream Amplifier Performance,” IEEE Transactions on Electron Devices, Vol. 37, No. 3, 1990, pp. 870-877. doi:10.1103/PhysRevA.36.2199
- A. K. Ganguly and H. P. Freund, “Three-Dimensional Nonlinear Simulation of a High-Power Collective Free-Electron Laser,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 331, No. 1-3, 1993, pp. 501-504. doi:10.1016/0168-9002(93)90097-2
- H. P. Freund and A. K. Ganguly, “Electron Orbits in FreeElectron Lasers with Helical Wiggler and Axial Guide Magnetic Fields,” IEEE Journal of Quantum Electronics, Vol. 21, No. 7, 1985, pp. 1073-1079. doi:10.1109/JQE.1985.1072758
- H. P. Freund and A. K. Ganguly, “High Efficiency Operation of Cherenkov Masers,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 304, No. 1-3, 1991, pp. 612-616. doi:10.1016/0168-9002(91)90938-M
- H. P. Freund and A. K. Ganguly, “Three-Dimensional Nonlinear Analysis of the Cherenkov Maser,” Nuclear Instruments and Methods in Physics Research Section A: Accelerators, Spectrometers, Detectors and Associated Equipment, Vol. 296, No. 1-3, 1990, pp. 462-467. doi:10.1016/0168-9002(91)90938-M

NOTES

^{*}Corresponding author.