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The following article has been retracted due to the investigation of complaints received against it. Mr. Mohammadali Ghorbani (corresponding author and also the last author) cheated the author’s name: Alireza Heidari. The scientific community takes a very strong view on this matter and we treat all unethical behavior such as plagiarism seriously. This paper published in Vol.3 No.3, 278-281, 2012, has been removed from this site.

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 [

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 [

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

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]:

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 [

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 [

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

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:

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:

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:

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

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

where and. Considering the oscillating commutation:

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

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

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

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

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.

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.