Ver 1.0

**DownLoad:** FELOde.zip

*Note: Click inside the applet and press the space bar to stop the simulation. Use
the tab key to clear the time graph.*

Many features of the dynamics of free electron lasers can be understood using a simple pendulum model. This model comes about because the electromagnetic wave (i.e., the laser field) and the magnetic field of the wiggler act in tandem on the electron to produce a sinusoidal potential similar to that of a pendulum in the earth's gravitational field,

|*A*|(1- cos(x+f)).

The constant *A *will depends on a number of FEL parameters including the distance
between the wiggler magnets, the strength of the wiggler field, and the energy of the
injected electrons. But most importantly, *A *is proportional to the laser field
strength. Imagine that an electron is part way up the potential well but falling toward
the potential minimum at theta =0. The energy released by the electron increases the laser
field and consequently lowers the minimum further. Conversely, electrons moving away from
the potential minimum up the potential well decrease the laser field. Since many electrons
are injected into the FEL simultaneously, the dynamics of the system can become very
complex indeed.

Both the equations of motion for individual electrons,

d^{2} x/dt^{2} = |*A*|
sin(x+f)

and the wave equation for the laser field,

d *A*/ dt = -J <exp(-*i *x)>

are solved by the FEL applet embedded in this document. The non-dimensional scaled
parameters, *A *and J, are proportional to the optical field strength and the current
density, respectively. The beam current density, J, determines the rate of change of the
laser field, A. The phase of the laser field phase, f, is of
course the phase of the complex scalar, *A*. The position of the electron is
determined by the pondermotive (or electron) phase, x. It is a
measure of the position of an electron with respect to the beat wave between the
electromagnetic and wiggler fields,

x=(k_{u}-k)z - wt.

The pendulum FEL model is valid for both weak or strong optical fields and for high or low gain. The theory does not include space charge effects, i.e., the electrons do not interact among themselves, nor does it include any off axis field dependence, i.e., the optical field is smoothly varying and one dimensional. The theory also assumes that the fractional energy change of the electron in passing through a single wiggler magnet is small.

Press the space bar to stop the applet before you change parameters.

**Small signal gain:**Stop the simulation and click on the reset tab. Set the initial field amplitude equal to zero and then click reset. Click on the time tab to view the field strength as a function of time. Notice that the field strength begins to increase exponentially. What is the time constant?**Saturation:**Stop the simulation and reset. Cick on the time tab let the simulation run untill the the field strength begins to decrease. (You can cheat and set the initial field amplitude to 1 if you have a show computer.) You have reached saturation. Reset the simulation and observe the time evolution in the phase space. Notice that the electrons initially fall into the ponermotive potential well. Notice the bunching of the electrons in the spacial distribution.**Field amplitude:**Continue to observe the time evolution of the field energy and the electron phase space. Notice that the field amplitude can both incearse and decrease but that it never again attains its first maximum value. The phase space shows that although the electrons begin to be distribute themself though phase space.**Single particle trajectories:**Set the number of particles equal to one and inject this electron into the FEL at different phases (with field amplitude equal one.) Observe both the time and phase space trajectories for this particle.**Beam Current:**Set the number of particles and the phase to one. Run the simulation at low and high beam currents. What is the effect on the separatrix?**Beam Energy:**Change the energy of the injected particles with amplitude and phase equal one.

Charles A. Brau, Free Electron Lasers, Academic Press, New York, 1990

W. B. Colson and A, M. Sessler, Ann. Rev. Nuc. Part. Sci., 1985 (35) 25-54

H. P. Freund and R. K. Parker, Free Electron Lasers in Encyclopedia of Physical Science and Technology 1991 Yearbook, Academic Press, New York 1991

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