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,
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,
d2 x/dt2 = |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=(ku-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.
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