Focusing properties of linear undulators

This paper investigates the focusing properties of linear magnetic undulators, i.e., devices characterized by weak defocusing properties in the horizontal (wiggling) plane and strongly focusing in the vertical plane. The problem of identifying the conditions that ensure the existence of the electron beam eigenstates in the undulator lattice for a given working point of electron beam energy $E_b$ and resonant wavelength ${\lambda }_r$ is studied. For any given undulator lattice, a bandlike structure is identified defining regions in the $(E_b,{\lambda }_r)$ plane where no periodic matching condition can be found, i.e., it is not possible to transport the electron beam so that optical functions are periodic at lattice boundaries. Some specific cases are discussed for the SPARC FEL undulator.

Figure 1

Layout of SPARC transfer line (quads $Q_1\to Q_6$) and undulator (quads $Q_7\to Q_{11}+\mathrm{\text{undulator sections}}U{M}_1\to U{M}_6$) of SPARC experiment. The $S$’s tag the positions of imaging screens, the $\zeta$’s measure the distance from photocathode.

Figure 2

Twiss “norm” [in the sense of Eqs. (6, 10, 11)] of lattice eigenstates in horizontal (red dashed line) and vertical (green solid line) directions as a function of the current in the quadrupole. The beam energy is $E=105\mathrm{MeV}$. The undulator is set at: (top) $g=10.88$ (${\lambda }_r=800\mathrm{nm}$); (bottom) $g=11.275\mathrm{mm}$ (${\lambda }_r=760\mathrm{nm}$). Positive current $I_Q$ means that the quadrupole is focusing in the vertical direction. Matching is possible whenever $\beta \gamma -{\alpha }^2=+1$ in both directions. No such interval exists in the first case, while in the latter the situation is reversed.

Figure 3

Top: maximum of Twiss norm [Eq. (6)] for lattice eigenstates in the vertical direction for the 1st undulator section as a function of ${\kappa }_{U,\mathrm{y}}$. The maximum is found by varying the quadrupole current in the range $-I_Q^{(x)}\le I_Q\le 0$. Dashed vertical lines delimit the boundaries of regions where no physical matching is possible. Bottom: ${\mathcal{N}}_{\max ,\mathrm{y}}$ as a function of the gap. Beam parameters are the same as in Fig. 2.

Figure 4

Top: resonant wavelength vs beam energy [Eq. (30)] corresponding to ${\kappa }_U=1.53,2.11,6.23\dots$ (see text). In the shaded regions it is impossible to realize physical vertical eigenstates with a single defocusing quadrupole. The black lines correspond to curves where the gap has been kept fixed to $g=8.2\mathrm{mm}$ (dashed), $g=11.5\mathrm{mm}$ (dash-dotted), and $g=20\mathrm{mm}$ (dotted), respectively. Bottom: the same as in the upper plot, zoomed to a region relevant for SPARC operation. The asterisks refer to the cases described in Fig. 2.

Figure 5

Top: merit functionals for undulator eigenstates corresponding to the quadrupole current (in abscissa) for a beam energy of 105 MeV ($\gamma \approx 205$) and gap values $g=11.944\mathrm{mm}$ (${\lambda }_r=700\mathrm{nm}$). Blue, magenta, orange, and red curves correspond to functionals (31, 32, 33), respectively; $2^{\mathrm{nd}}\to 3^{\mathrm{rd}}$ row: optical functions ${\beta }_x$ (blue) and ${\beta }_y$ (magenta) for eigenstates corresponding to the minima of ${\chi }^2$ functionals (31, 32) (the crosses atop blue and magenta curves in the top plot).

Figure 6

Characteristic behavior of horizontal (continuous line) and vertical eigenstates (dashed line) belonging to 1st to 3rd allowed region in Fig. 4 (top to bottom). The $\beta$’s correspond to optimum according to merit functional (34). While the horizontal pattern remains substantially unchanged, the vertical eigenstate exhibits a number of peaks equal to the order of the allowed band to which it belongs.

Figure 7

RADIA model of SPARC undulator.

Figure 8

Parameter $\delta$ defined in (a3) vs the gap for the SPARC undulator, modeled with RADIA (fit function $f=a+b*g$ with $a=-9.11\times {10}^{-3}$, $b=1.18\times {10}^{-3}{\mathrm{mm}}^{-1}$).