Appendix D — Auto-phasing Algorithm

D.1 Standing Wave Cavity

For external RF fields in OPAL-T, the instantaneous field is the interpolated spatial field multiplied by cos(omega t + phi). The energy gain of a particle is therefore

\[ \Delta E(\varphi, r) = q V_0 \int_{z_{\mathrm{begin}}}^{z_{\mathrm{end}}} \cos(\omega t(z,\varphi) + \varphi)\, E_z(z,r)\, dz. \tag{D.1}\]

To maximize the gain, one differentiates with respect to the lag \varphi and sets the derivative to zero. This yields the lag rule

\[ \tan(\varphi) = - \Gamma_1 / \Gamma_2. \tag{D.2}\]

The original appendix defines

\[ \Gamma_1 = \sum_{i=1}^{N-1} \left(1 + \omega \frac{\partial t}{\partial\varphi}\right) \int_{z_{i-1}}^{z_i} \sin\!\left(\omega \left(t_{i-1} + \Delta t_i\frac{z-z_{i-1}}{\Delta z_i}\right)\right) \left(E_{z,i-1} + \Delta E_{z,i}\frac{z-z_{i-1}}{\Delta z_i}\right) dz, \tag{D.3}\]

and

\[ \Gamma_2 = \sum_{i=1}^{N-1} \left(1 + \omega \frac{\partial t}{\partial\varphi}\right) \int_{z_{i-1}}^{z_i} \cos\!\left(\omega \left(t_{i-1} + \Delta t_i\frac{z-z_{i-1}}{\Delta z_i}\right)\right) \left(E_{z,i-1} + \Delta E_{z,i}\frac{z-z_{i-1}}{\Delta z_i}\right) dz. \tag{D.4}\]

Assuming piecewise-linear field and time-of-flight models between samples, the integrals can be evaluated analytically in terms of four coefficients:

\[ \Gamma_{11,i} = -\frac{\cos(\omega t_i) - \cos(\omega t_{i-1})}{\omega\Delta t_i}, \qquad \Gamma_{12,i} = \frac{-\omega\Delta t_i\cos(\omega t_i) + \sin(\omega t_i) - \sin(\omega t_{i-1})}{\omega^2 (\Delta t_i)^2}, \tag{D.5}\]

\[ \Gamma_{21,i} = \frac{\sin(\omega t_i) - \sin(\omega t_{i-1})}{\omega\Delta t_i}, \qquad \Gamma_{22,i} = \frac{\omega\Delta t_i\sin(\omega t_i) + \cos(\omega t_i) - \cos(\omega t_{i-1})}{\omega^2 (\Delta t_i)^2}. \tag{D.6}\]

With these, the appendix rewrites the accumulated sums in a form suitable for an iterative implementation.

K[i] = K[i-1] + (z[i] - z[0]) * q * V;
b[i] = sqrt(1. - 1. / ((K[i] - K[i-1]) / (2.*m*c^2) + 1)^2);
t[i] = t[0] + (z[i] - z[0]) / (c * b[i])

D.2 Traveling Wave Structure

Traveling-wave auto-phasing is slightly more complicated because the field is composed of:

• an entry fringe field
• two standing-wave core contributions with phase offsets
• an exit fringe field

The appendix uses the field map FINLB02-RAC.T7 as the representative traveling-wave example.

The corresponding energy gain is written as the sum of four integrals over the entry fringe, two core components, and exit fringe. In the notation of the appendix, the two core contributions are shifted by one cell length s and by phase offsets \varphi_{c1} and \varphi_{c2}.

FINLB02_RAC: TravelingWave, L=2.80, VOLT=14.750*30/31,
             NUMCELLS=40, FMAPFN="FINLB02-RAC.T7",
             ELEMEDGE=2.67066, MODE=1/3,
             FREQ=1498.956, LAG=FINLB02_RAC_lag;