Optics Analysis¶

This page summarizes how the optics analysis is performed with our omc3 software, from the physics point of view. Please note that this page is meant as a gentle introduction including references rather than a comprehensive treatment of the topics therein.

A practical walkthrough

To see how to use the omc3 package to do so, refer to the omc3 analysis workflow page.

All quantities described here are reconstructed from the turn-by-turn (TbT) centroid positions recorded by BPMs around the ring following beam excitation. The preceding spectral analysis step, in which the tune, amplitude and phase of spectral lines are extracted from TbT data, is documented in the harmonic analysis page.

Linear Optics¶

Phase Advances¶

The betatron phase \(\phi_{x,y}(s)\) at the curvilinear coordinate \(s\) is defined by the integral:

\[ \phi_{x,y}(s) = \int_0^s \frac{ds}{\beta_{x,y}(s)} . \]

The phase advance corresponds to the difference of the betatron phase functions at two points, typically also taken with respect to an arbitrary initial point at \(s = 0\):

\[ \phi_{s_1 \rightarrow s_2} = \phi(s_{2}) - \phi(s_{1}) = \int_{s_{1}}^{s_{2}} \frac{1}{\beta(s)} ds . \]

The tune \(Q_{x,y}\) is the total phase advance per revolution, and given \(C\) the machine circumference is written as:

\[ Q_{x,y} = \frac{1}{2 \pi} \Delta \phi_{x,y} = \frac{1}{2 \pi} \oint_C \dfrac{ds}{\beta_{x,y} (s)} , \]

The phase advance between two BPMs is extracted from TbT data as the difference of the spectral line phases at the tune frequency at the two locations.

In the N-BPM method (see Beta from Phase), the relevant inputs are phase advances between non-consecutive BPMs. Combinations with phase separations well away from \(0\) and \(\pi\) are preferred, as these minimise the sensitivity of the cotangent terms to measurement noise. Specifically, \(\pi/2\) and \(3 \pi/2\) are ideal.

Special phases

The special phases correspond to the phase advances between specific machine elements of interest, usually the AC Dipole kicker magnet to tertiary collimators in the IRs for the LHC.

Action¶

The Courant-Snyder action \(J_{x,y}\) is the conserved invariant of free betatron motion, related to the oscillation amplitude by \(J_{x,y} = A_{x,y}^2(s)/(2\beta_{x,y}(s))\) at any location \(s\). Since it cannot be read off from a single BPM without knowing \(\beta_{x,y}(s)\), a calibration-dependent estimate is formed by averaging over \(N\) BPMs:

\[ 2J_{x,y} = \frac{1}{N} \sum_{n=1}^N \frac{\left(\text{peak-to-peak}/2\right)_n^2}{\beta_{x,y}^m(s_n)} . \]

For an AC dipole excitation, \(J_{x,y}\) is modulated during the ramp-up and ramp-down phases but is constant on the flat-top plateau; only plateau turns enter the analysis.

Beta from Amplitude¶

The \(\beta\)-function can be estimated from the oscillation amplitude \(A_{x,y}\) recorded at each BPM. From the Courant-Snyder parameterisation, the oscillation amplitude is \(A_{x,y}(s) = \sqrt{2J_{x,y} \beta_{x,y}(s)}\), giving:

\[ \beta_{x,y}^\text{amp}(s_i) = \frac{A_{x,y}^2(s_i)}{2 J_{x,y}} . \]

Because the action \(J\) must itself be estimated from the peak-to-peak amplitudes and model \(\beta\)-functions (see above), this method is sensitive to BPM calibration errors and is generally less accurate than \(\beta\) from phase. It is used as a cross-check and as a diagnostic for BPM calibration.

Beta from Phase¶

The \(\beta\)-function at BPM \(i\) can be determined from the measured phase advances to two other BPMs \(j\) and \(k\), with normalisation using model values. The three-BPM combination formula reads:

\[ \beta_{x,y}^\text{phase}(s_i) = \frac{\cot\!\left(\phi_{x,y}(i \to j)\right) + \cot\!\left(\phi_{x,y}(i \to k)\right)} {\cot\!\left(\phi_{x,y}^m(i \to j)\right) + \cot\!\left(\phi_{x,y}^m(i \to k)\right)} \beta_{x,y}^m(s_i) , \]

where superscript \(m\) denotes model values.

The analytical N-BPM method

The analytical N-BPM method (Wegscheider et al., Phys. Rev. Accel. Beams 20, 111002, 2017) extends this calculation by averaging over \(N\) specifically chosen BPM combinations which remove unfavorable phase advances. The method also includes the known statistical uncertainties of various elements for error estimation.

When the beam is driven by an AC dipole, the measured driven beta functions differ from the free ones since the AC dipole modifies the parametrization of the particle coordinates. Details on this effect can be found in F. Soubelet's PhD Thesis. This effect can be compensated analytically (see information for instance in the output files meaning section's admonition).

Beta-Beating¶

The very commonly looked at \(\beta\)-beating, the deviation from model values, goes as:

\[ \frac{\Delta\beta_{x,y}(s)}{\beta_{x,y}(s)} = \frac{\beta_{x,y}^\text{phase}(s) - \beta_{x,y}^m(s)}{\beta_{x,y}^m(s)} . \]

It is a primary value of interest for the quantification of the optics' quality throughout the machine.

Dispersion and Normalized Dispersion¶

The dispersion function \(D_{x,y}(s)\) quantifies the sensitivity of the closed orbit to a relative momentum offset \(\delta = \Delta p / p_0\):

\[ z_\text{co}(s,\,\delta) = z_{\text{co},0}(s) + D_z(s)\,\delta + \mathcal{O}(\delta^2) , z = x, y , \]

To determine dispersion in practice, \(\delta\) is varied by adjusting the RF frequency away from its nominal value, which shifts the beam energy. The resulting mean orbit change at each BPM, plotted versus \(\delta\), yields \(D_{x,y}(s)\) as the slope.

3D Excitation

A more efficient approach is to perform measurements with AC dipole excitation in which the RF frequency is simultaneously modulated, providing an "excitation" of the three degrees of freedom at once. This method was explored but is not currently actively used.

The normalized dispersion, written as \(D_{x,y} / \sqrt{\beta_{x,y}}\), is independent of BPM calibration factors, making it a more robust observable of the sensitivity to energy deviations.

Coupling¶

Linear betatron coupling mixes the motions of horizontal and vertical planes. It is parameterised by the resonance driving terms \(f_{1001}\) and \(f_{1010}\).

These are reconstructed from the cross-plane spectral lines of the same TbT data used for the linear optics. A dedicated page on reconstructing the coupling terms is available.

Nonlinear Optics¶

Resonance Driving Terms¶

Nonlinear multipole errors in the lattice excite resonance lines in the betatron spectrum. From normal form theory, the RDT \(f_{jklm}\), associated with resonance \((j-k, l-m)\), drives a spectral line in the horizontal TbT spectrum at frequency \(f_\text{res} = (j - k)\,Q_x + (l - m)\,Q_y\).

Some References

An explanation of the emergence of the \(f_{jklm}\) terms from nonlinearities' treatment via normal forms can be found in the PhD theses of F. Carlier and F. Soubelet, including references. A complete parametrization of the RDT terms is found in the paper by R. Tomás. This detailed paper by Franchi et al. provides tables of spectral lines to RDTs correspondence.

The complex value of the RDT at each BPM is encoded in the amplitude and phase of the complex Courant-Snyder spectral line \(H^+(j{-}k,\, l{-}m)\) at this frequency.

Combined Resonance Driving Terms¶

The combined RDTs, denoted with capital \(F\) terms, are particular linear combinations of existing RDTs. A table is provided in the paper by Franchi et al..

For instance, the combined coupling RDTs \(F_{xy}\) and \(F_{yx}\) are expressed as:

\[ \begin{align} F_{xy} &= \frac{\sinh 2 \mathcal{P}}{\mathcal{P}} \bigl( f_{1001} - f_{1010}^* \bigr) , \\ F_{yx} &= \frac{\sinh 2 \mathcal{P}}{\mathcal{P}} \bigl( f_{1001}^* + f_{1010}^* \bigr) , \end{align} \]

where \(2\mathcal{P} = \sqrt{|2f_{1010}|^2 - |2f_{1001}|^2}\) and \({}^*\) denotes complex conjugation. Their practical advantage is that they can be reconstructed directly from transverse position coordinates alone. The impact of the combined coupling RDTs into position parametrization is found in this paper by M. Hofer.

Chromatic Coupling¶

Chromatic coupling describes the variation of the global coupling quantity \(|C^{-}|\) with momentum offset \(\delta\). It arises from sextupolar errors in the magnets in combination with dispersion. An off-momentum particle will experience a skew quadrupolar field from a sextupole in dispersive regions (vertical dispersion and normal sextupole, or horizontal dispersion and skew sextupole). Since there is larger horizontal dispersion in a ring, skew sextupolar components are the dominant source of chromatic coupling.

It is measured by repeating optics measurements at several RF frequency settings (each corresponding to a different \(\delta\)) and fitting the resulting \(|C^{-}|\) as a polynomial in \(\delta\). It has been defined by the coefficient:

\[ \left| \frac{d C^{-}}{d\delta} \right| \approx 4 \Delta Q \left| \frac{d f_{1001}}{d \delta} \right| \]

A study of chromatic coupling in the LHC and its correction can be found in this paper by Persson et al..