Single-turn tune estimation in coupled systems
In circular accelerators, single-particle tunes can be estimated from turn-by-turn data using the Fast Fourier Transform (FFT). If we only have a few turns, we can estimate the tunes using the Average Phase Advance (APA) method [1]. Using APA, we can estimate the tunes from the phase space coordiantes on two neighboring turns.
Tune-estimation in uncoupled systems
In a 1D system (2D phase space), we normalize the \(x-x'\) phase space coordinates such that the turn-by-turn normalized coordiantes \(u_x-u_x'\) jump around a circle of area \(A_x = 2 \pi J_x = \pi (u_x^2 + u_x'^2)\). Here \(J_x\) is the Courant-Snyder (CS) invariant, or “action”. The phase angle \(\theta_x\) is defined by
\[ \tan(\theta_x) = u_x' / u_x. \tag{1}\]
The tune \(\nu_x\) is estimated from the average phase advance over \(N\) turns:
\[ \nu_x = \frac{1}{2 \pi N} \sum_{t=1}^{N} \left( \theta_x^{(t)} - \theta_x^{(t - 1)} \right) \tag{2}\]
The vertical pane is treated in the same way. The normalized coordinates \(u_x, u_x'\) are defined as [2]:
\[ \begin{align} \tilde{x} &= x - \eta_x \delta_p, \\ \tilde{x}' &= x' - \eta_x' \delta_p, \\ \begin{bmatrix} u_x \\ u_x' \end{bmatrix} &= \sqrt{\frac{1}{\beta}} \begin{bmatrix} 1 & 0 \\ \alpha_x & \beta_x \\ \end{bmatrix} \begin{bmatrix} \tilde{x} \\ \tilde{x}' \end{bmatrix}, \end{align} \tag{3}\]
where \(\delta_p = (p - p_0) / p_0\) is the fractional momentum offset and \(\\{ \alpha_x, \beta_x, \eta_x \\}\) are the CS parameters and dispersion computed from the transfer matrix.
Extension to coupled systems
To extend this analysis to \(N\)-dimensional systems (\(2N\)-dimensional phase space), we can define a \(2N \times 2N\) normalization matrix \(\mathbf{V}^{-1}\) [3]:
\[ \mathbf{u} = \mathbf{V}^{-1} \mathbf{x}, \tag{4}\]
with phase space coordinates \(\mathbf{x}\) and normalized coordiantes \(\mathbf{u}\):
\[ \begin{align} \mathbf{x} &= [x_1, x_1', \dots, x_N, x_N' ]^T, \\ \mathbf{u} &= [u_1, u_1', \dots, u_N, u_N' ]^T. \end{align} \tag{5}\]
If the matrix is designed correctly, it will convert the one-turn transfer matrix \(\mathbf{M}\) to the form:
\[ \mathbf{M} = \mathbf{V} \mathbf{P} \mathbf{V}^{-1}, \tag{6}\]
where the \(\mathbf{P}\) is a block-diagonal phase advance matrix:
\[ \begin{aligned} \mathbf{P}(\nu_1, \nu_2, \nu_3) &= \begin{bmatrix} \mathbf{R}(2 \pi \nu_1) & \mathbf{0} & \mathbf{0} \\ \mathbf{0} & \mathbf{R}(2 \pi \nu_2) & \mathbf{0} \\ \mathbf{0} & \mathbf{0} & \mathbf{R}(2 \pi \nu_3) \end{bmatrix} \\\ \mathbf{R}(\theta) &= \begin{bmatrix} +\cos\theta & +\sin\theta \\\ -\sin\theta & +\cos\theta \end{bmatrix} \end{aligned} \tag{7}\]
The phase \(\phi_k\) in mode \(k\) is computed as in the 2D analysis (\(\tan(\theta_k) = u_k' / u_k\)). The tune \(\nu_k\) is again estimated from the average phase advance:
\[ \nu_k = \frac{1}{2 \pi N} \sum_{t=1}^{N} \left( \theta_k^{(t)} - \theta_k^{(t - 1)} \right), \tag{8}\]
And the action is given by:
\[ J_{k} = \frac{u_k^2 + u_k'^2}{2}. \tag{9}\]
Designing the normalization matrix
In a linear lattice, we can compute the \(\mathbf{V}\) matrix using the eigenvectors of the transfer matrix.
\[ \mathbf{M} \mathbf{v}_k = \lambda_k \mathbf{v}_k. \tag{10}\]
For an \(N\)-dimensional system, the eigenvectors come in \(N\) pairs: \(\{ \mathbf{v}_k, \mathbf{v}_k^* \}\), where \(^*\) means complex conjugate. The \(\mathbf{V}\) matrix then takes the form [2]:
\[ \mathbf{V} = [Re(\mathbf{v}_1), -Im(\mathbf{v}_1), Re(\mathbf{v}_2), -Im(\mathbf{v}_2), \dots]. \tag{11}\]
The tunes \(\nu_k\) are related to the eigenvalues.
\[ \lambda_k = \exp(-2 \pi i \nu_k) \tag{12}\]
Alternatively, if we have a collection of particles and know that the bunch’s covariance matrix \(\mathbf{\Sigma} = \langle \mathbf{x} \mathbf{x}^T \rangle\) is “matched” to the transfer matrix (\(\mathbf{M}\mathbf{\Sigma}\mathbf{M}^T = \mathbf{\Sigma}\)), we can compute the eigenvectors without the transfer matrix by solving:
\[ \mathbf{\Sigma} \mathbf{U} \mathbf{v} = i\varepsilon_k \mathbf{v}. \tag{13}\]
Here, \(\varepsilon_k\) are the invariant emittances (“eigenemittances”, “intrinsic emittances”, “mode emittances”), whose product is the total emittance:
\[ \varepsilon \equiv | \mathbf{\Sigma} |^{1/2} = \prod_{k} \varepsilon_k. \tag{14}\]
The \(\mathbf{V}\) matrix is then constructed from the eigenvectors in the same way as above.