Betatron coupling: Merging Hamiltonian and matrix approaches

Betatron coupling is usually analyzed using either matrix formalism or Hamiltonian perturbation theory. The latter is less exact but provides a better physical insight. In this paper direct relations are derived between the two formalisms. This makes it possible to interpret the matrix approach in terms of resonances, as well as use results of both formalisms indistinctly. An approach to measure the complete coupling matrix and its determinant from turn-by-turn data is presented. Simulations using methodical accelerator design MAD-X, an accelerator design and tracking program, were performed to validate the relations and understand the scope of their application to real accelerators such as the Relativistic Heavy Ion Collider.

Figure 1

Top: $4|f_{1001}|$ and $|[({\overline{C}}_{12}-{\overline{C}}_{21})+i({\overline{C}}_{11}+{\overline{C}}_{22})]/\gamma |$ plotted as a function of the longitudinal position. Bottom: $|\overline{\mathbf{C}}|/{\gamma }^2$ and $4(|f_{1001}{|}^2-|f_{1010}{|}^2)$ plotted as a function of longitudinal position. Horizontal and vertical tunes are $Q_x=18.226$ and $Q_y=17.257$, respectively.

Figure 2

Top: Mean of the ratio of $|[({\overline{C}}_{12}-{\overline{C}}_{21})+i({\overline{C}}_{11}+{\overline{C}}_{22})]/\gamma |$ and $4|f_{1001}|$. Bottom: Mean of the ratio of $|\overline{\mathbf{C}}|/{\gamma }^2$ and $4(|f_{1001}{|}^2-|f_{1010}{|}^2)$. The error bars represent the standard deviation of the ratio of the respective quantities. The horizontal and vertical tunes are $Q_x=18.226$ and $Q_y=17.232$.

Figure 3

$|f_{1001}{|}^2$, $|f_{1010}{|}^2$, and $|\overline{\mathbf{C}}|/{\gamma }^2$ plotted as a function of $Q_x$ ($Q_y=0.228$). The stop bands at the sum and the difference resonance show the dramatic increase in these functions. The $1/2$ integer resonance line is plotted as a reference.

Figure 4

Top: Comparison of $|C|/{\gamma }^2$ between MAD-X model and SVD computed values from tracking data. Bottom: Difference between model and calculated values of $|C|/{\gamma }^2$. Horizontal and vertical tunes are $Q_x=18.226$ and $Q_y=17.257$, respectively.

Figure 5

Normalized rms of the difference of $|C|/{\gamma }^2$ between MAD-X model and SVD computed values for increasing amount of Gaussian noise in turn-by-turn data. Horizontal and vertical tunes are $Q_x=18.226$ and $Q_y=17.257$, respectively.

Figure 6

Top: Comparison of $|\overline{\mathbf{C}}|/{\gamma }^2$ between MAD-X model and SVD computed values from tracking data. Middle: Difference between model and calculated values of $|\overline{\mathbf{C}}|/{\gamma }^2$. Bottom: A representation of the lattice (dipoles in black and quadrupoles in red) is shown in the bottom graph. The horizontal and vertical tunes were $Q_x=28.266$ and $Q_y=29.212$, respectively, and $\Delta Q_{\min }=4.37\times {10}^{-2}$.

Figure 7

Schematic view of a skew quad and the neighbor BPMs.

Figure 8

Top: Skew quadrupole strengths calculated from RDTs and $\overline{\mathbf{C}}$ matrix are compared to MAD-X model values. Note that RDTs are calculated from $\overline{\mathbf{C}}$ matrix using Eqs. (9, 10). Middle: $|\overline{\mathbf{C}}|/{\gamma }^2$ is plotted as a function longitudinal position. A representation of the lattice (dipoles in black and quadrupoles in red) is shown in the bottom graph. The horizontal and vertical tunes are $Q_x=28.266$ and $Q_y=29.212$, respectively, and $\Delta Q_{\min }=4.37\times {10}^{-2}$.

Figure 9

Top: $\Delta Q_{\min }$ calculated using the three different values of RDTs at respective locations of skew quadrupoles as a function of their strength. Bottom: $\Delta Q_{\min }$ calculated using three different values of $|\overline{\mathbf{C}}|$ at respective locations of skew quadrupoles as a function of their strength ($Q_x=18.226$, $Q_y=17.239$).