Figure 1

Coordinate systems in a BPM. The laboratory coordinate system is defined with respect to the survey positions. BPM alignment and electronics errors mean that the coordinates returned by the BPM are with respect to the “BPM axes” which are not perfectly aligned with the laboratory system. Coupling means that resonant excitation of the transverse modes leads to motion along axes different from the laboratory or BPM axes. The four BPM buttons are labeled according to the convention used in CESR.Reuse & Permissions

Figure 2

Lattice functions in CesrTA. Top: horizontal (black) and vertical (red) beta functions. Bottom: horizontal dispersion, with BPM locations indicated by blue circles, and skew quadrupole locations indicated by vertical dashed lines.Reuse & Permissions

Figure 3

Simulation of turn-by-turn button signals on a BPM (7W). Each plot shows a correlation between two BPM button signals or beam coordinates (returned by the BPM), obtained by tracking a bunch performing coherent oscillations in normal mode I. An ideal model (with no errors) of the CesrTA lattice is used: tracking data are shown as blue points; fitted ellipses are shown in red; the major axes of the ellipses are shown as black lines. The ellipses are more clearly visible when coupling is present (see Fig. 4). Button signals are calculated using Eq. (21), with relative gain errors applied to the buttons as follows: $g_2/g_1=1.063$, $g_3/g_1=1.282$, $g_4/g_1=1.170$. The button locations are given by $x_0=20\mathrm{mm}$, $y_0=10\mathrm{mm}$. $x$ and $y$ beam coordinates are calculated from Eqs. (23, 24), but using nominal gain values $g_1=g_2=g_3=g_4$.Reuse & Permissions

Figure 4

As Fig. 3, but with skew quadrupole 48W turned on with strength $k=-0.023{\mathrm{m}}^{-2}$.Reuse & Permissions

Figure 5

Emittance estimated from dispersion in CesrTA with (top) 8.1 MV rf voltage, and (bottom) 0.81 MV rf voltage. The lines show the distribution (over 1000 seeds of machine errors) of the ratio of the estimated mode II emittance to the actual mode II emittance; the vertical axes are scaled so that each curve has unit area. In the normal mode case, the mode II dispersion is used, and the emittance estimated using Eq. (10). In the Cartesian case, a modified version of Eq. (10) is used, in which the mode II dispersion is replaced by the vertical dispersion in Eq. (12). In both cases, the exact emittance is calculated using the envelope method. The approximations involved in Eq. (10) are valid for small values of the synchrotron tune (which is 0.074 at 8.1 MV rf, and 0.023 at 0.81 MV rf).Reuse & Permissions

Figure 6

Distribution (from 1000 seeds of machine errors with rms shown in Table ) of final mode II emittance (top and bottom, left) and dispersion (top and bottom, right) after correction of dispersion measured at the BPMs. The vertical axes are scaled so that each curve has unit area. Top: emittance (left) and vertical dispersion (right) after tuning based on correction of vertical ($y$) dispersion. Bottom: emittance (left) and mode II dispersion (right) after tuning based on correction of mode II dispersion. In each case, the dispersion is the rms of the dispersion measured at the BPMs (including measurement errors, corresponding to a resolution of 12 mm). Note the different horizontal scales between the top plots and the bottom plots. Note also that the emittance distribution for the correction based on vertical dispersion is sensitive to the magnitude of the BPM gain errors, whereas the emittance distribution for a correction based on mode II dispersion is insensitive to the BPM gain errors. Black lines correspond to no BPM gain errors; blue lines correspond to 2% (vertical dispersion correction) or 5% (normal mode dispersion correction) gain errors; and red lines correspond to 5% (vertical dispersion correction) or 10% (normal mode dispersion correction) gain errors.Reuse & Permissions

Figure 7

Correlation plots for the button signals from BPM 7W during resonant excitation of normal mode I. Turn-by-turn signals from the BPM buttons (and the coordinates calculated using the nominal calibration) are shown as blue points. The points in each correlation plot are fitted with a red ellipse; the major axis of the ellipse is shown with a black line. The ellipses described by the data points are more obvious if the coupling is increased: compare with Fig. 12. Compare also with the simulation results shown in Fig. 3.Reuse & Permissions

Figure 8

Fourier spectra of beam position coordinates returned by BPM 7W over 1024 consecutive turns, with mode I resonant excitation. Note that the (fractional part of the) mode I tune is 0.59, and the (fractional part of the) mode II tune is 0.63.Reuse & Permissions

Figure 9

Fourier spectra of beam position coordinates returned by BPM 7W over 1024 consecutive turns, with mode II resonant excitation. Note that the (fractional part of the) mode I tune is 0.59, and the (fractional part of the) mode II tune is 0.63.Reuse & Permissions

Figure 10

Measured mode II dispersion after correction of orbit, (vertical) dispersion, and coupling. The middle plot shows a comparison between the measured dispersion, and the dispersion in a model with skew quadrupole strengths adjusted to fit the measurement. The final skew quadrupole strengths found from the fitting are shown in the top plot (the order of the quadrupoles is the same as shown in Fig. 2); the bottom plot shows the residual between the measurement and the fitted model. The rms is the root mean square over the measurements at the BPMs.Reuse & Permissions

Figure 11

Effect of change in strength of skew quadrupole 48W on mode II dispersion in CesrTA. Top: skew quadrupole strengths in a model fitted to the measured mode II dispersion. Center: comparison between measured mode II dispersion and the mode II dispersion in the fitted model. Bottom: residual between measured and fitted mode II dispersion. The rms is the root mean square over the measurements at the BPMs.Reuse & Permissions

Figure 12

Correlation plots for the button signals from BPM 7W during resonant excitation of normal mode I, with the strength of skew quadrupole 48W changed by $\Delta k=-0.023{\mathrm{m}}^{-2}$. The blue points show the data from the BPM; in each correlation plot, the points are fitted with a red ellipse; the major axis of each ellipse is shown with a black line. Compare with the simulation results shown in Fig. 4.Reuse & Permissions

Figure 13

Variation in skew quadrupole strengths in a model fitted to the measured mode II dispersion, over a range of applied changes in strength of skew quadrupole 48W (SKQ48W). The fitted versus applied strength of skew quadrupole 48W (black line, with data points indicated by circles) shows a linear behavior, with gradient close to 1 (shown by a dashed line). Other skew quadrupoles (colored lines) show some (approximately) linear response, but with much smaller gradients.Reuse & Permissions

Figure 14

Results of fitting a model to a measurement of the mode II dispersion. The measurement was made after all skew quadrupoles were turned off, following an initial low-emittance tuning using conventional techniques (vertical orbit and dispersion correction, and coupling correction). Top: fitted skew quadrupole strengths. Middle: measured and fitted mode II dispersion. Bottom: residual between measured and fitted mode II dispersion. Note that in this case, a poor fit is achieved in the model to the mode II dispersion. This suggests that the skew quadrupoles would not be effective in correcting the real sources of mode II dispersion in the machine.Reuse & Permissions

Figure 15

Results from simulation of iterating a mode II dispersion correction for tuning the mode II emittance. Top: with a scaling factor of 1 applied to the computed changes in skew quadrupole strengths, there are significant fluctuations in the mode II emittance in successive iteration. Bottom: with a scaling factor of 0.4, the fluctuations are smoothed out, and the emittance (and dispersion) converge to some final values. The same magnitude of machine errors was applied as used in the simulations described in Sec. 3d. Note that the same seed of random errors was used to generate both top and bottom plots.Reuse & Permissions