Harmonics undulator

Figure 1. The distribution of distances between an undulating membrane and a rigid surface interacting via a harmonic potential is the same as the Boltzmann distribution of distances between a rigid surface and independent planar pieces of a suitable chosen area A, interacting via the same potential.

Let us first summarize the results obtained for a thermal undulating membrane interacting with a rigid, flat surface via a harmonic potential, U f u) = (1/2)Bit2. The total energy, per unit area, is given by20... 

There is only one case in which the thermal undulations of a membrane can be evaluated analytically, namely when one membrane undulates in a harmonic potential. Let us assume that the position, along the z direction, of each point of the membrane is described by the function u(x,y), with u(x,y) = 0 corresponding to a non undulating membrane (see Fig. B.l). 

Prazeres R, Glotin F, Jaroszynski DA, Ortega JM, Rippon C (1999) Enhancement of harmonic generation using a two section undulator. Nucl Instrum Methods Phys Res Sect A-Accel Spectrom Dect Assoc Equip 429 131-135... 

Fig. 5. Calculated variation of the brilliance of the undulator fundamental and third harmonics as a function of the energy upon tuning [22]...

The simplest optics making use of the full undulator peak is a pinhole camera without focusing elements. The only optical element necessary is a plane mirror in order to elevate the beam out of the plane of the storage ring, to reduce the background due to Bremsstrahlung and to cut the harmonics (Fig. 10 b). 

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