Application of transfer matrix and transfer function analysis to grating-type dielectric laser accelerators: Ponderomotive focusing of electrons

The question of suitability of transfer matrix description of electrons traversing grating-type dielectric laser acceleration (DLA) structures is addressed. It is shown that although matrix considerations lead to interesting insights, the basic transfer properties of DLA cells cannot be described by a matrix. A more general notion of a transfer function is shown to be a simple and useful tool for formulating problems of particle dynamics in DLA. As an example, a focusing structure is proposed which works simultaneously for all electron phases.

Figure 1

The principle of FODO focusing. The best known realization of this principle is the alternating gradient focusing used in conventional rf accelerators.

Figure 2

A segment of a grating-type DLA. The unit cell is iterated in one dimension along the electron beam.

Figure 3

Example parameters used for calculation, based on Ref. [15].

Figure 4

The properties of the transfer function for the unit cell shown in Fig. 3. (a) Assumed $(x_1,x_1^{\prime })$ at the entrance of the cell. (b) Calculated $(x_2,x_2^{\prime })$ at the exit of the cell. The result depends on the initial phase ${\mathrm{\Phi }}_1$ (color coded). The thicker lines correspond to six selected phases: $-\pi ,-\frac{2}{3}\pi ,-\frac{1}{3}\pi ,0,\frac{1}{3}\pi ,\frac{2}{3}\pi$.

Figure 5

The transfer properties of simple optical elements: (a) The assumed set of input pairs $(x_1,x_1^{\prime })$, representing incident rays with various positions and zero slopes. (b)–(f) Calculated $(x_2,x_2^{\prime })$ at the exit of (b) converging lens, (c) diverging lens, (d) linear prism, (e) nonlinear prism, (f) converging lens with off-axis focus. Linear transfer functions describe (a)–(c), for (d), (f) an affine function is needed, while for (e) a nonlinear transfer function must be used.

Figure 6

A DLA accelerator segment analogous to a FODO structure.

Figure 7

The focusing properties of the accelerator segment shown in Fig. 6, calculated for the unit cell shown in Fig. 3 using Eq. (a1). (a) Assumed $(x_1,x_1^{\prime })$ at the entrance of the segment: a parallel beam. (b) Calculated $(x_2,x_2^{\prime })$ at the exit of the segment, for $n=1$ and $m=0$. (c) Calculated $(x_2,x_2^{\prime })$ at the exit of the segment, for $n=8$ and $m=5$.

Figure 8

Boundary field effects can be handled using boundary cells with corresponding ${\mathcal{B}}_{\pm }$ transfer functions.

Figure 9

Alternating lens (FODO) (a), and alternating nonlinear prisms (b), are examples of ponderomotive focusing systems.

Figure 10

The focusing properties of the ${\mathcal{O}}_{5+1/2}{\mathcal{R}}^8{\mathcal{O}}_{5+1/2}{\mathcal{R}}^8$ segment calculated using “spoiled” (${\delta }_2={\delta }_1$) transfer function [compare with the correct result in Fig. 7].

Figure 11

(a) Time dependence of the focusing force in a FODO-like DLA cell (Fig. 6). (b) Harmonic oscillation leading to classical ponderomotive force.

Figure 12

Equation (d4), based on the transfer properties of an elementary DLA cell, plotted for the same six phases as in Fig. 7.