Undulator helical field

Helical undulators build on this principle by using two orthogonal magnetic field arrays [82, 83]. These permit transverse excursions in perpendicular x and y directions. If the arrays have a relative longitudinal shift, this introduces a phase to the induced perpendicular excursions and when the phase is 90° the electron trajectory can follow left- or right-handed corkscrew paths The emitted radiation is correspondingly right- or left-handed CPL. 

Let both the helical axis and the electric field are parallel to the normal z of a cholesteric liquid crystal layer of thickness d and >0. In the case of a very weak field the elastic forces tend to preserve the original stack-like arrangement of the cholesteric quasi-layers as shown in Fig. 12.15a. On the contrary, in a very strong field, the dielectric torque causes the local directors to be parallel to the cell normal, as shown in Fig. 12.15c. At intermediate fields, due to competition of the elastic and electric forces an undulation pattern appears pictured in Fig. 12.15b. Such a structure has two wavevectors, one along the z-axis (nld) and the other along the arbitrary direction x within the xy-plane. The periodicity of the director pattern results in periodicity in the distribution of the refractive index. Hence, a diffraction grating forms. Let us find a threshold field for this instability. 

Fig. 12.15 A planar cholesteric structure in the electric field parallel to the helical axis (fi > 0). The local director oiicaitaticm is shown by solid lines field-off planar alignment (a), undulated structure in a weak field > E> E,h (b), and the homeotropic structure in the field exceeding the threshold fin helix tmwinding E > E (c)...

---