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K-Modulation

This section gives a brief overview over the K-Modulation method. A more detailed description can be found in M. Minty and F. Zimmermann's book1 and the references therein.

Also available on this site is a checklist for conducting K-Modulation measurements in the LHC.

The full K-Modulation analysis is two-fold: The K-Modulation GUI is used for LHC measurements, and the following analysis is part of the omc3 package.

K-Modulation is a complementary optics measurement method which consists in changing the gradient of a quadrupole and measuring the induced tune variation. The average \(\beta\)-function in the modulated quadrupole is linked to the gradient change \(\Delta K\) and tune change \(\Delta Q_{x,y}\) via1:

\[ \beta_{x,y} = \pm 2 \Delta K^{-1}\Big[ cot(2 \pi Q_{x,y}) [ 1 - cos(2 \pi \Delta Q_{x,y}) ] + sin(2 \pi \Delta Q_{x,y}) \Big] \]

If these measurements are conducted for two adjacent quadrupoles, the evolution of the \(\beta\)-function in-between the modulated quadrupoles can also be inferred2. Here, the average \(\beta\)-function in the quadrupole is expressed in terms of the optics functions \(\beta_0\), \(\alpha_0\), and \(\gamma_0\) at the end of the quadrupole.

Assuming a drift space between the quadrupoles, these coordinates can then be expressed in terms of the distance of the quadrupole end to the middle of the drift-section \(L^*\), the minimum \(\beta\)-function \(\beta^*\), and \(w\), the offset of this minimum with respect to the center of the drift. The length \(L^*\) is usually obtained from the machine layout. Using the two average \(\beta\)-functions in the quadrupoles, the other two variables \(\beta^*\) and \(w\) can then be calculated. The \(\beta\)-function at other elements in the drift space can then be determined by propagation.

Compared to other methods, K-Modulation allows to infer a potential waist shift and its direction, which is not possible using the turn-by-turn based methods. However, K-Modulation is usually more time-intensive, and is only applicable with individually powered quadrupoles.

  1. Measurement and Control of Charged Particle Beams, Michiko G. Minty, Frank Zimmermann, https://doi.org/10.1007/978-3-662-08581-3 
    @book{Minty:629879,
  author = {Minty, Michiko G and Zimmermann, Frank},
  title = {Measurement and control of charged particle beams},
  publisher = {Springer},
  address = {Berlin},
  series = {Particle acceleration and detection},
  year = {2003},
  url = {https://cds.cern.ch/record/629879},
  doi = {10.1007/978-3-662-08581-3}
}

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  2. Accuracy and Feasibility of the Beta* Measurement for LHC and High Luminosity LHC Using K Modulation, F. Carlier, and R. Tomás, Phys. Rev. Accel. Beams 20, 2017 
    @article{PhysRevAccelBeams.20.011005,
  title = {Accuracy and Feasibility of the Beta* Measurement for {{LHC}} and {{High Luminosity LHC}} Using k Modulation},
  author = {Carlier, F. and Tom{\'a}s, R.},
  year = {2017},
  month = jan,
  volume = {20},
  pages = {011005},
  doi = {10.1103/PhysRevAccelBeams.20.011005},
  url = {https://link.aps.org/doi/10.1103/PhysRevAccelBeams.20.011005},
  journal = {Phys. Rev. Accel. Beams},
  keywords = {read},
  number = {1}
}

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