Analytical considerations for linear and nonlinear optimization of the theoretical minimum emittance cells: Application to the Compact Linear Collider predamping rings

The theoretical minimum emittance cells are the optimal configurations for achieving the absolute minimum emittance, if specific optics constraints are satisfied at the middle of the cell’s dipole. Linear lattice design options based on an analytical approach for the theoretical minimum emittance cells are presented in this paper. In particular the parametrization of the quadrupole strengths and optics functions with respect to the emittance and drift lengths is derived. A multiparametric space can be then created with all the cell parameters, from which one can choose any of them to be optimized. An application of this approach is finally presented for the linear and nonlinear optimization of the Compact Linear Collider predamping rings.

Figure 1

Schematic layout of the TME cell.

Figure 2

Left: Parametrization of the vertical phase advance of a TME cell with a dipole bending angle of $\theta =2\pi /19$ with the drift spaces lengths $s_1$, $s_2$, $s_3$, when targeting the theoretical minimum emittance. Only solutions providing optical stability are presented. Right: The stability region in the ($s_1,s_2$) plane for different theoretical minimum emittance targets (${\epsilon }_{\text{xTME}}\sim N_d^{-3}$), color-coded with $s_3$.

Figure 3

Parametrization of the quadrupole focal lengths (top), $f_1$ (left) and $f_2$ (right) and the horizontal (bottom, left) and vertical (bottom, right) chromaticities with the drift spaces lengths, $s_1$, $s_2$ providing optical stability.

Figure 4

Left: The ($s_1$, $s_2$, $s_3$) combinations that provide the lowest chromaticity in both planes (${\xi }_{x,y}\le -2$), color-coded with the total length of the cell. Right: Projection of the low chromaticity solutions onto the ($s_1$, $s_2$) plane.

Figure 5

Parametrization of the relative horizontal beta ${\beta }_r$ and relative dispersion $D_r$ at the center of the dipole (left) and the quadrupole focal lengths (right), with the cell detuning factor ${\epsilon }_r$. The stable solutions are indicated with the black squares, while the stable and feasible ones with the magenta triangles.

Figure 6

Parametrization of the cell detuning factor ${\epsilon }_r$ (left), and the horizontal (middle) and vertical (right) chromaticities with the horizontal and vertical phase advances of the cell is presented for each case, for a conventional (top) and a modified (bottom) TME cell.

Figure 7

Comparison between the analytical solution and madx simulations. The analytical solutions are presented in black, the solutions satisfying the stability requirements in red and the results from madx for different quadrupole lengths: $l_q=1\mathrm{cm}$ in green, $l_q=10\mathrm{cm}$ in blue, and $l_q=20\mathrm{cm}$ in ciel.

Figure 8

Parametrization of the horizontal (left) and vertical (right) chromaticity with the drift spaces lengths $s_1$ and $s_2$ and the horizontal dispersion at the middle of the dipole, for a detuned cell (${\epsilon }_r=10$).

Figure 9

Horizontal and vertical phase advances of the PDR TME cell, parametrized with the third order Hamiltonian amplitudes.

Figure 10

Horizontal and vertical phase advances of the PDR TME cell parametrized with the fourth order Hamiltonian amplitudes.

Figure 11

Top: Parametrization of the horizontal (top, left) and vertical (top, right) amplitude dependent tune shift, with the horizontal and vertical phase advances of the TME cell. Bottom: Parametrization of the square root of the quadratic sum of the horizontal and vertical amplitude dependent tune shifts (left) and of the fifth radiation integral (right), with the horizontal and vertical phase advances of the TME cell.

Figure 12

The fifth order resonance driving terms for which $|j-k|=1$ and $|l-m|=4$.

Figure 13

The optical functions of the TME arc cell of the PDR.

Figure 14

The working point in tune space for $\delta p/p$ from $-1.2$ to 1.2% (blue) and the first order tune shift with amplitude up to $6\sigma$ (green). The on-momentum working point is (16.39, 12.27).

Figure 15

The required acceptance around the PDR in order to fit the positron beam in units meters (left) and in units of beam sizes (right).

Figure 16

The on and off momentum DA of the PDR for ${\delta }_p=0$ (red), 1.2% (green) and $-1.2\%$ (blue).

Figure 17

Diffusion maps (left) and frequency maps (right) for $\delta p/p=1.2\%$ (top), 0 (middle), and $-1.2$ (bottom).