Measuring ${\beta }^{*}$ in SuperKEKB with $K$ modulation

SuperKEKB is an asymmetric energy electron positrion collider currently under commissioning in Japan. It aims to achieve a record high luminosity of $8\times {10}^{35}{\mathrm{cm}}^{-2}{\mathrm{s}}^{-1}$, for which accurate control of ${\beta }_y^{*}$ is needed. The advanced final focus system is also relevant for studies related to future linear colliders, which similarly need to achieve very small beam sizes. To work for SuperKEKB, the $K$ modulation technique is generalized to allow known quadrupole fields between the modulated magnets and the interaction point. Initial measurements taken in HER agree with simulations, and show that $K$ modulation is suitable for measuring the minimum of the $\beta$ function at the interaction point within 1%, however it is too uncertain to be used for measuring the displacement of the beam waist away from the interaction point. In addition, tune shift equations for a modulated quadrupole are derived without assuming a thin lens perturbation, giving a simple method to calculate the tune shift to second order in the quadrupole strength modulation.

Figure 1

Measured vertical beam sizes in ATF2 [17] for different values of ${\beta }_x^{*}$ and ${\beta }_y^{*}$ as compared to the nominal value. On the left are the nominal vertical beam sizes intended for SuperKEKB rings HER and LER. An additional line is added for the case of reduced ${\beta }_y^{*}$ in LER.

Figure 2

Illustration of a one turn matrix (top left), and a one turn matrix with a thin quadrupole perturbation $\mathrm{\Delta }K$ (top right). Below is a one turn matrix where a quadrupole of strength $K$ is removed (bottom left), and a one turn matrix where a quadrupole of strength $K$ has been replaced by a quadrupole of strength $K+\mathrm{\Delta }K$ (bottom right).

Figure 3

The minimum of the $\beta$ function is ${\beta }_w$ and it is displaced longitudinally away from the IP a distance $w$. The $\beta$ function grows quadratically away from the waist. $w$ is defined as positive when the waist lies downstream from the IP.

Figure 4

Illustration of several hard edge quadrupole magnets, together modeling a quadrupole with varying field. The slices are numbered, ${\beta }_0$, ${\alpha }_0$, and ${\gamma }_0$ corresponds to the beta function at the entrance of slice 0.

Figure 5

Quadrupole fields in the orbits through the IR of LER (above) and HER (below). The names of the magnets are indicated. Fields from the inner quadrupoles in LER that leak into the HER orbit are pointed out with red rectangles. The focusing system is nearly symmetric, but notice slightly different lengths and placements of horizontally focusing magnets in HER.

Figure 6

Deviation of the calculated ${\beta }_y^{*}$ as a function of ${\beta }_y^{*}$ in LER, found from simulations using sad in the absence of uncertainties.

Figure 7

Deviations in the calculated ${\beta }_y^{*}$ for different values of coupling strength in LER. The simulations are done for ${\beta }_y^{*}$ values of $500\mu \mathrm{m}$, $270\mu \mathrm{m}$ and $180\mu \mathrm{m}$. The quadrupole modulations are done such that the tune shifts away from the difference coupling resonance. The tune separation is 0.04.

Figure 8

Comparison of tune shifts for different ${\beta }_y^{*}$ calculated using Eq. (3) and taking into account the fringes by using Eq. (34). The calculations are based on the values for the LER magnet QC1LP, with $\mathrm{\Delta }KL/KL=2\times {10}^{-5}$.

Figure 9

Uncertainty in the ${\beta }_w$ measurement in LER for different uncertainties in the tune measurement, as well as for different values of modulation strength $\mathrm{\Delta }KL$. Simulated for nominal ${\beta }_y^{*}=270\mu \mathrm{m}$.

Figure 10

Uncertainty in the $w$ measurement in LER for different uncertainties in the tune measurement, as well as for different values of modulation strength $\mathrm{\Delta }KL$. Simulated for nominal ${\beta }_y^{*}=270\mu \mathrm{m}$.

Figure 11

Simulated and measured tune shift when modulating the inner quadrupoles in HER. The simulations are done using the design optics with ${\beta }_y^{*}=3\mathrm{mm}$. The uncertainty in the recorded tune measurements are $4\times {10}^{-4}$ for the modulation of the QC1LE magnet and $2\times {10}^{-4}$ for the modulation of the QC1RE magnet. The lines have been made by fitting the average $\beta$ function and using Eq. (3). The integrated strength of the magnets during the measurements were $K{L}_0=-1.15{\mathrm{m}}^{-1}$ for QC1LE and $K{L}_0=-1.09{\mathrm{m}}^{-1}$ for QC1RE.

Figure 12

Histograms showing the number of BPMs measuring a certain value of the tune. Three sets of turn by turn measurements were recorded for each setting of the two magnets QC1LE and QC1RE. The colors indicate the sequence of the measurements; blue being first, followed by orange, and green last. The histograms on the left column come from the reference measurements for the magnets with $\mathrm{\Delta }KL=0$, while in the ones on the right column have $\mathrm{\Delta }KL/KL=0.05\%$. The integrated strength of the magnets during the measurements were $K{L}_0=-1.15{\mathrm{m}}^{-1}$ for QC1LE and $K{L}_0=-1.09{\mathrm{m}}^{-1}$ for QC1RE.

Figure 13

Simulated and measured tune shift when modulating the inner quadrupoles in HER. The simulations are done using ${\beta }_y^{*}=3.2\mathrm{mm}$. The uncertainty in the recorded tune measurements are $4\times {10}^{-4}$ for the modulation of the QC1LE magnet and $2\times {10}^{-4}$ for the modulation of the QC1RE magnet. The lines have been made by fitting the average $\beta$ function and using Eq. (3). The integrated strength of the magnets during the measurements were $K{L}_0=-1.15{\mathrm{m}}^{-1}$ for QC1LE and $K{L}_0=-1.09{\mathrm{m}}^{-1}$ for QC1RE.

Figure 14

The spectrum of the vertical position measurements from one BPM in HER during the $K$ modulation measurements. The main frequency components of the turn by turn (TbT) data correspond to the horizontal and vertical tunes in the ring shown in tune units.

Figure 15

Coupling in HER during the $K$ modulation measurements, estimated from the spectrum of turn by turn data from all BPMs [24]. When modulating the QC1LE magnet by $\mathrm{\Delta }KL/KL=-0.05\%$ the estimated coupling was a lot larger $|C^{-}|=0.07$, this measurement also shows the largest change in the orbit, see Fig. 16.

Figure 16

Orbits calculated as the average of all position measurements from turn by turn data taken during the $K$ modulation measurements. The different colors indicate different modulations in the QC1LE and QC1RE magnets, left and right of the IP, respectively. The orbits are plotted as the deviation from the orbits found before modulating the magnets.

Figure 17

Measured vertical and horizontal tunes while modulating the quadrupole magnets closest to the IP in HER, QC1LE and QC1RE. The smallest separation between the horizontal and vertical tunes is approximately 0.04, which is 20 times larger than the estimated coupling strength during the measurements, $|C^{-}|\approx 0.002$, see Fig. 15. The integrated strength of the magnets during the measurements were $K{L}_0=-1.15{\mathrm{m}}^{-1}$ for QC1LE and $K{L}_0=-1.09{\mathrm{m}}^{-1}$ for QC1RE.

Figure 18

Measured ${\beta }_y^{*}$ as a function of $\mathrm{\Delta }KL$ for the $K$ modulation measurements in HER. Error bars of 1.5% are included.