Alternative Beam-Based Alignment techniques for the SIRIUS Storage Ring

Masters Qualification Exam

Vitor Davi de Souza

Prof. Dr. Tulio Costa Rizuti da Rocha

May 15, 2026

Contents

    Introduction

    SIRIUS

    SIRIUS is the Brazilian 4th-generation synchrotron light source, operated by the Brazilian Synchrotron Light Laboratory (LNLS) at the Brazilian Center for Research in Energy and Materials (CNPEM) in Campinas, São Paulo.

    SIRIUS Storage Ring Parameters
    Energy 3 GeV
    Emittance (hor.) 250 pm.rad
    Operation mode Top-up
    Parameter Currently Phase II
    Beam current 200 mA 350 mA
    Lifetime ≈ 17 h ≈ 10 h
    Number of beamlines A B
    SIRIUS

    Synchrotron Light Sources: Storage Rings

    Dipoles
    Quadrupoles
    Sextupoles
    Vacuum Chambers
    RF Cavities
    Beam Position Monitors
    Injection
    System

    Motivation

    • The SIRIUS's Storage Ring needs sub-micrometer orbit stability to maintain its performance
    • The orbit stability depends on the precision and accuracy of the Beam Position Monitors (BPM)
    • The BPMs have high precision (nanometer-level) but low accuracy
    • So we calibrate the BPMs using the Beam-Based Alignment (BBA) procedure

    The Problem

    The current BBA procedure is very time-consuming (6∼8 hours)

    Objectives

    • Study and validate alternative Beam-Based Alignment techniques for the SIRIUS Storage Ring

    Bibliographical Review

    • Standard BBA
    • Parallel BBA
    • Fast AC BBA
    • Others

    Simulations

    Test BBA methods with the SIRIUS Storage Ring model

    Experiments

    Implement and validate alternative BBA methods

    Theoretical Background

    Coordinate System

    We adopt a curvilinear coordinate system that follows a design orbit $\vec{r}_0(s)$ (the Frenet-Serret frame)

    $\vec{v}$
    $\vec{r}_0$
    $\hat{x}$
    $\hat{y}$
    $\hat{s}$
    • $\hat{s}(s) = \dfrac{d}{ds}\vec{r}_0(s)$
    • $\hat{x}(s) = -\rho(s)\dfrac{d}{ds}\hat{s}(s)$
    • $\hat{y}(s) = \hat{s}(s) \times \hat{x}(s)$

    For high energy particles ($v \rightarrow c$), their state is described in terms of small deviations from the design orbit $$\mathbf{x} = (x, x', y, y')^T$$

    Magnetic Fields

    The transverse magnetic fields are also described in the Frenet-Serret frame

    We normalize the field coefficients by the magnetic rigidity $(ec/E_0)$, defining the magnetic lattice functions

    Dipole:
    $$\Biggl\{ \begin{array}{l} B_x = 0 \\ B_y = B_0 \end{array} \Biggr.$$
    Dipole:

    $G(s) \equiv \dfrac{ec}{E_0}B_0(s) = \dfrac{1}{\rho(s)}$

    (Curvature function)

    Quadrupole:
    $$\Biggl\{ \begin{array}{l} B_x = B_1 y \\ B_y = B_1 x \end{array} \Biggr.$$
    Quadrupole:

    $K(s) \equiv \dfrac{ec}{E_0}B_1(s)$

    (Focusing function)

    Skew Quadrupole:
    $$\Biggl\{ \begin{array}{l} B_x = A_1 x\\ B_y = -A_1 y \end{array} \Biggr.$$
    Skew Quadrupole:

    $K_S(s) \equiv \dfrac{ec}{E_0}A_1(s)$

    (Skew quad. strength)

    Transverse Dynamics

    In the linear approximation, the equations of motion are

    $$\Biggl\{ \begin{array}{l} x'' = -\left(K + G^2\right)x + G\delta + K_S y \\ y'' = + K y + K_S x \end{array} \Biggr.$$

    where $\delta = \frac{p - p_0}{p_0} \approx \frac{E - E_0}{E_0}$

    The solutions for $x$ and $y$ can be expressed with a matrix formalism $$\mathbf{x}(s_1) = M_{s_1 \leftarrow s_0}\, \mathbf{x}(s_0)$$

    Drift

    $ \begin{pmatrix} x \\ x' \end{pmatrix}_{\!\!s_1} = \begin{pmatrix} 1 & L \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ x' \end{pmatrix}_{\!\!s_0} $

    Quadrupole

    $ \begin{pmatrix} x \\ x' \end{pmatrix}_{\!\!s_1} = \begin{pmatrix} 1 & 0 \\ -KL & 1 \end{pmatrix} \begin{pmatrix} x \\ x' \end{pmatrix}_{\!\!s_0} $

    Corrector

    $ \begin{pmatrix} x \\ x' \end{pmatrix}_{\!\!s_1} = \begin{pmatrix} 1 & 0 \\ 0 & 1 \end{pmatrix} \begin{pmatrix} x \\ x' \end{pmatrix}_{\!\!s_0} + \begin{pmatrix} 0 \\ \theta \end{pmatrix} $

    Closed Orbit

    The passage through all elements of the accelerator is described by the one-turn matrix $$M = M_{(s_0 + L) \leftarrow s_0} = M_{L \leftarrow s_{N}} \cdots \cdot M_{s_2 \leftarrow s_1} \cdot M_{s_1 \leftarrow s_0}$$

    The closed orbit is the fixed point of the one-turn matrix $$\mathbf{x}^*(s) = M \mathbf{x}^*(s)$$

    We often call closed orbit the $x$ and $y$ components of $\mathbf{x}^*(s)$ $$\vec{u} = \{x^*_{s_0}, x^*_{s_1}, \dots, x^*_{s_{N}}, y^*_{s_0}, y^*_{s_1}, \dots, y^*_{s_{N}}\}$$

    In an ideal accelerator, the closed orbit is the design orbit

    Orbit Distortion

    However, dipolar kicks disturb the closed orbit $$\frac{ec}{E_0}\Delta B = \Delta x' = \theta \quad \Rightarrow \quad \mathbf{x}^*(s_0) = M \mathbf{x}^*(s_0) + \begin{pmatrix} 0 \\ \theta \end{pmatrix} $$

    The disturbance is propagated and particles now follow the perturbed closed orbit

    The main sources of orbit distortion are:

    • Field imperfections
    • Misalignments
    • Corrector magnets

    Orbit Correction

    For the accelerator to perform as designed, we need the closed orbit the closest to the design orbit

    →To do this, we need to observe and correct the orbit

    • BPMs measure the orbit and the orbit distortion ($\Delta\vec{u}$)

    • Corrector magnets kick the beam ($\Delta\vec{\theta}$)

    • With the Orbit Response Matrix ($\mathbf{M}$) we know how the kicks affect the orbit

    $$\Delta\vec{u} = \mathbf{M}\Delta\vec{\theta}$$

    Then, to correct the orbit, we minimize $$\left|\Delta\vec{u} - \mathbf{M}\Delta\vec{\theta}\right|^2$$

    → Which provides the necessary kicks to correct the orbit $$\Delta\vec{\theta} = - \mathbf{M}^\dagger\Delta\vec{u}$$

    Orbit Feedback-System: Beam Position Monitors

    • At SIRIUS: 160 BPMs

    Two pairs of antennas that read the transverse beam position

    $$x \propto \dfrac{(A + D) - (B + C)}{A + B + C + D}$$ $$y \propto \dfrac{(A + B) - (C + D)}{A + B + C + D}$$

    $$x \propto \dfrac{{\color{blue}(A + D)} - \color{red}{(B + C)}}{A + B + C + D}$$ $$y \propto \dfrac{{\color{green}(A + B)} - {\color{darkorange}(C + D)}}{A + B + C + D}$$

    → Sub-micron resolution

    Orbit Feedback-System: Correctors

    At SIRIUS:

    • 120 CH (slow horizontal correctors)
    • 160 CV (slow vertical correctors)
    • 80 FCH (fast horizontal correctors)
    • 80 FCV (fast vertical correctors)

    Orbit Feedback System: SOFB and FOFB

    SOFB

    • Slow Orbit FeedBack
    • @ 1 Hz
    • 160 BPMs
    • 120 CH + 160 CV
    • Compensates slow drifts
      • Thermal effects
      • Ground motion
      • Residual misalignments

    FOFB

    • Fast Orbit FeedBack
    • @ 48 kHz
    • 80 BPMs
    • 80 FCH + 80 FCV
    • Compensates disturbances from
      0.1 Hz to 1 kHz

    SOFB download kicks (4% @ 10 Hz) from FOFB to avoid the saturation of the fast correctors

    Orbit Feedback System: the accuracy problem

    The electric center of BPMs does not necessarily match the design orbit, and general misalignments also make the BPMs inaccurate

    → To solve this we use the beam as probe to calibrate the BPMs
    (Beam-Based Alignment techniques)

    Beam-Based Alignment

    The BBA Principle

    Standard BBA

    The standard BBA method emerged in the 80s∼90s with seminal works

    • R. Schmidt, "Misalignments from K-modulation" (CERN SL/93-19, 1993)
    • P. Röjsel, "A Beam Position Measurement System Using Quadrupole Magnets Magnetic Centra as the Position Reference" (NIM, A 343, 1994)
    • Barnett et al., "Dynamic Beam Based Alignment" (AIP 333, 1995)

    The method empirically applies the BBA principle by modulating the strength of quadrupoles and scanning the orbit on the BPMs to find the offsets

    → In the SIRIUS Storage Ring, the method was studied and implemented in 2019–2020 (commissioning)

    Standard BBA

    The standard BBA procedure consists of:

    • → Scan the beam position on a BPM: $r_i$ = $x_i$ or $y_i$
    •      → For each $r_i$, vary the quadrupole strength: $K^\pm = K_0 \pm \Delta K$
    •           → For each $K^\pm$, collect the orbit on all BPMs: $\vec{u}^\pm_i$
    •                → Calculate the orbit distortion: $\Delta \vec{u}_i$
    • → Minimize $\left|\Delta u - (ar + b)\right|^2$ (linear fit) or $\left|(\Delta u)^2 - (ar^2 + br + c)\right|$ (quadratic fit)
    • → Obtain the offsets: $r_\text{offset} = \left\langle\dfrac{-b}{a}\right\rangle$ or $r_\text{offset} = \left(\dfrac{-b}{2a}\right)$

    Standard BBA

    At SIRIUS:

    • 160 BPMs
    • 90 normal quadrupoles and 70 skew quadrupoles
    • orbit shifts on both planes ($x$, $y$) simultaneously
    • 9 scan points per BPM (range: ±100μm)
    • approximately 2∼3 min per BPM
    • 6∼8 hours for a full BBA procedure

    Standard BBA

    Typical standard BBA results at SIRIUS for a single BPM

    Parallel BBA

    The Parallel Beam Beam Alignment (PBBA) was proposed by Xiaobiao Huang in 2022

    Since then, it has been simulated and experimented on accelerators such as

    • LCLS-II (USA, SLAC, 2022)
    • SPEAR3 (USA, SLAC, 2022)
    • FCC-ee (Switzerland, CERN, 2023/2024)
    • KARA (Germany, KIT, 2024/2025)
    • NSLS-II (USA, BNL, 2024)

    Parallel BBA

    • The central idea is to minimize the Induced Orbit Shift (IOS)

    → Our goal is to find an orbit that satisfies: $$\vec{u}(\vec{\theta}, \vec{K}_0) = \vec{u}(\vec{\theta}, \vec{K}^\pm)$$

    → The IOS is defined as: $$\vec{\xi} = \frac{1}{2}\left(\vec{u}(\vec{\theta}, \vec{K}^+) - \vec{u}(\vec{\theta}, \vec{K}^-)\right)$$

    → Minimizing the IOS results in: $$\vec{\xi} = 0 \Leftrightarrow \vec{u}(\vec{\theta}, \vec{K}^+) = \vec{u}(\vec{\theta}, \vec{K}^-) \Leftrightarrow \vec{u}(\vec{\theta}, \vec{K}_0) = \vec{u}(\vec{\theta}, \vec{K}^\pm)$$

    → Linearizing $\vec{\xi}$ around an initial kick configuration $\vec{\theta}_0$ gives: $$\vec{\xi}\ \ =\ \ \vec{\xi}(\vec{\theta_0}) + \dfrac{\partial \vec{\xi}}{\partial \vec{\theta}}(\vec{\theta} - \vec{\theta_0}) \ \ =\ \ \vec{\xi}_0 + \mathbf{R}\Delta\vec{\theta}$$

    where $\mathbf{R}_{ij} = \partial \vec{\xi}_i/\partial \vec{\theta}_j$ is the IOS Response Matrix

    → Solving for $\vec{\xi} = 0$ we get: $$\Delta\vec{\theta} = -\mathbf{R}^{-1}\vec{\xi}_0$$

    $\vec{\theta} = \vec{\theta_0} + \Delta\vec{\theta} \quad\Rightarrow\quad \vec{u}_\text{offset} = \vec{u}(\vec{\theta}, \vec{K}_0) = \vec{u}(\vec{\theta}, \vec{K}^\pm)$

    Parallel BBA

    The Parallel BBA procedure consists of:

    • $\rightarrow$ Separate all BPM-Quad pairs into groups
    • $\rightarrow$ Define $\vec{K}^+$ and $\vec{K}^-$ for each group
    • $\rightarrow$ Get the IOS Response Matrix $\mathbf{R}$ for each group
    • $\rightarrow$ For each group, do:
    • $\qquad\rightarrow$ Loop while $\vec{\xi} \neq 0$
    • $\qquad\qquad\rightarrow$ Measure the IOS: $\vec{\xi} = \frac{1}{2}\left(\vec{u}(\vec{\theta}, K^+) - \vec{u}(\vec{\theta}, K^-)\right)$
    • $\qquad\qquad\rightarrow$ Compute the delta kicks ($\Delta\vec{\theta} = -\mathbf{R}^{-1}\vec{\xi})$ and apply ($\vec{\theta} = \vec{\theta} + \Delta\vec{\theta}$)
    • $\qquad\rightarrow$ Read and save the offsets: $\vec{u}_\text{offset} = \vec{u}(\vec{\theta}, \vec{K}_0) = \vec{u}(\vec{\theta}, \vec{K}^\pm)$

    Preliminary Results

    Parallel BBA: Simulations

    → First steps: divide BPM-Quad pairs into groups

    Q4, QDB2, QDP2, QS @ (M2, C3)   Q1, QS @ (C1, C3)   QS, Q1 @ (C1, C4)   QS, QDB2, QDP2 @ (M1, C2)

    Parallel BBA: Simulations

    → Analyze the impact of measuring the IOS

    Tunes and coupling variation during IOS measurement

    Parallel BBA: Simulations

    → Simulate a PBBA run on the SIRIUS Storage Ring model

    PBBA: Group 0 (Q4, QDB2, QDP2, QS @ [M2, C3])

    Parallel BBA: Experiments

    • Oct 13, 2025 — First experiments (Groups 0 & 1 and a group of selected BPMs)
    • Dec 8, 2025 — First full PBBA runs
    • Jan 5, 2026 — Parallel BBA vs Standard BBA
    • Jan 12 and Feb 9, 2026 — Groups rearrangement and convegence analysis

    Parallel BBA: Experiments

    • October 13, 2025

    PBBA and Standard BBA for a group of 4 selected BPMs

    PBBA for the "default" group 0

    • The Standard BBA runs for the group of 4 selected BPMs took ∼8 minutes (≈ 2min/BPM)
    • Each PBBA run for both groups took ∼1 minute → does not scale with the number of BPMs

    • For the group of 4 selected BPMs, the results between the Standard BBA and PBBA differ by ±2μm

    • For the "default" group 0: the results of each PBBA run differ by ±6μm

    Parallel BBA: Experiments

    • December 8, 2025: Full PBBA (all "default" groups)

    • Each full PBBA run took ≈ 4min (1 min/group)

    • For all groups, the differences between each PBBA run are smaller then ±8μm

    Parallel BBA: Experiments

    • January 5, 2025: Full PBBA before and after a full Standard BBA

    • The differences between the results of Standard BBA and PBBA are smaller then ±30μm
    with $\sigma_x$ = ±3.46μm and $\sigma_y$ = ±6.78μm

    Parallel BBA: Experiments

    • January 12 and February 9, 2026

    • For all the configuration of groups, the convegence is achieved in 3∼4 iterations
    → which reduces the run execution time by 33%∼50%

    • The configuration of 8 groups is the most stable: kicks converge and stay still

    • For a single BPM, the differences between the Standard BBA and PBBA runs are:
    Groups 4x40: ($x$) ±6μm and ($y$) ±7μm
    Groups 8x20: ($x$) ±2.5μm and ($y$) ±4μm
    Groups 16x10: ($x$) ±5μm and ($y$) ±7μm

    • The Standard BBA results from the linear and quadratic fits differ by ($x$) ±0.5μm and ($y$) ±3μm

    Parallel BBA: Partial Conclusion

    Until now PBBA has shown:

    • High speed — the execution time is 45∼90 times faster (6 hours to 4∼8 minutes)
    • Accuracy — comparisons with Standard BBA show differences of a few microns
    • Repeatability — PBBA runs under the same configuration differ by a few microns

    Closing

    So far, the following activities have been developed:

    • Study of accelerator physics, beam dynamics, beam-based alignment
    • Implementation and experiments of Parallel BBA on the SIRIUS Storage Ring
    • Courses: FI001, FI195, FI189 (and FI008 to be concluded this semester)

    Next Steps

    • A contribution work to be presented at the IPAC'26 (International Particle Accelerator Conference)
    • Continue PBBA experiments to improve and polish the implementation
    • Study the FBBA-AC method in greater depth
    • Implement and validate FBBA-AC at SIRIUS
    • Compare the overall performance of the different methods
    • Survey of additional methods

    Thank you!

    vitor.souza@lnls.br