1.2.1 Enlarging the Available Installation Length for the VSR Cryomodule

Due to increasing interest in studies that require very short photon pulses, HZB is developing a new operational mode called Variable pulse-length Storage Ring (VSR) [4], which aims to fulfill the requirements of the users who need short 2 ps bunches and of the users relying on the high average beam current, which is mainly stored in the 15 ps long bunches. The idea is to store long and short bunches in one storage ring simultaneously by establishing a beating pattern of higher harmonic cavities providing alternating longitudinal focusing gradients for two different bunch lengths.

This will be achieved by the installation of an additional cavity module consisting of two 1.5 GHz and two 1.75 GHz cavities. The superposition of these two new frequencies and the existing 0.5 GHz cavity leads to alternating high (blue) and low gradients (blue) shown in Figure 1.2. The long bunches are located at the small voltage gradients, where the voltages of the 1.5 GHz and 1.75 GHz cavities cancel out. The short bunches, with higher current than in the low alpha mode, are produced at the high voltage gradient, where the voltages of the cavities add up.

The additional cavity module will be installed in the T2 section of the BESSY II storage ring, shown in Figure 1.3. One challenge - addressed by my bachelor’s thesis [5] - is that the module needs more space than initially assumed. One solution is to remove the two Q5T2 quadrupoles to gain about 0.7 meters of installation length. The goal is to switch off the Q5T2 quadrupoles in simulations while maintaining the most important transverse linear parameters as the beta functions, tune, and momentum compaction factor. The best optics presented showed that a turn-off of the Q5T2 quadrupoles in the T2 straight is possible. Furthermore, the optics was tested at the storage ring, where it was possible to store high current with reasonable lifetime and injection efficiency.

As already stated in the conclusion of the bachelor’s thesis, the presented optics have to be further optimized regarding different aspects: There is still a non-negligible beta beat outside of the T2 section. The linear optics could be further improved by overcoming some limitations of the code of the bachelor’s thesis. Depending on the number of parameters, some optimization runs took several days. Optimizing time-critical parts of the developed code would allow optimizing using a higher number of quadrupole configurations. Also, the capability to mask certain regions, which would not be taking into account by the objective function, or set the beta functions to the desired value in the cavity module could further improve the obtained optics. That would require a significant rewrite of the developed code. Adjusting the objective function of the optimization method, discussed in Section 4.1, could improve the linear result. Besides the optimization of the linear beam dynamics, the optics has to be further optimized with regard to the non-linear dynamics. Different sextupole settings can be used to optimize the phase and momentum acceptance.

1.2.2 Emittance Exchange Experiment

With the uprise of 4ᵗʰ generation light sources, the discussion of round beams becomes more relevant. A round beam is produced when the emittance is the same in both transversal planes. Such an emittance exchange is helpful in two use cases:

First, discussed in [6], an exchange of the transverse beam emittances can improve the injection efficiency. Since the beam is injected in the horizontal plane and the beam emittance is much larger in the horizontal plane for an electron synchrotron, the horizontal emittance primarily defines the required acceptance for a high injection efficiency. Therefore, reducing the horizontal emittance in the booster synchrotron, shown in Figure 1.4, by partially transferring it into the vertical plane can significantly lower the required acceptance of the storage ring and thus increase the injection efficiency.

Another reason for an emittance exchange is the generation of round beams in the storage ring. The goal is to match the electron beam phase space with the phase space of the emitted photon beam at the radiation source to optimize the photon beam’s brilliance and coherent fraction.

There are different techniques to generate round beams. For example, operating on a coupling resonance or driving a skew excitation resonantly with the beam. One experiment to study the influence of the resonant skew excitation on the emittance exchange depending on the size of the beta function required an optics change in one of the triplet sections of the storage ring. Section 4.2 presents how the developed code was used to raise the horizontal beta function from 1.2 m to 15 m at the center of the straight.

1.2.3 Automated Lattice Summaries

At HZB, the development of the Conceptual Design Report (CDR) of the BESSY II successor BESSY III has started. Many lattice candidates arise during the development of an entirely new accelerator lattice, which themselves require serval iterations. These numerous versions of lattices are created by different physicists making different considerations and optimizing for different parameters. Due to the variety of accelerator physics codes, the lattice files and simulation results are stored in different formats. Even though many accelerator codes provide the option to export the lattice files to different formats, this often does not work flawlessly and requires editing these files manually afterward. That makes sharing and comparing the different lattices candidates during the design phase very cumbersome and unnecessarily time-consuming.

The number of different simulation codes and lattice file formats is probably due to the variety of different lattice development tasks. Certain tasks can be very specific to a given facility and require a feature, which may not be implemented in one of the existing codes. Due to some codes not being open-source or since it can be easier to implement a certain feature from scratch than integrating it into one of the existing codes, this probably led to the historical fragmentation of accelerator physics codes and lattice file formats.

On the other hand, the boundary conditions for a new BESSY III lattice are well defined. Therefore, especially for fundamental lattices development, mainly linear beam dynamics, it should be possible to automate the process of sharing lattice files and simulation results.

Ideally, there would be a shared database of lattices. The lattices files would be automatically generated in multiple formats every time a lattice is added to the database. In the same way, a predefined set of simulations would be run. The simulation results could then be summarized in the form of a lattice report, for example, through a web page. Such a framework of automatically generated lattice summaries would also be useful for existing lattices. For example, to benchmark a BESSY III candidate lattice with one of the existing 4ᵗʰ generation light sources or build an updated version of the Synchrotron Light Source Data Book [7].

As a starting point and to facilitate the creation of such a database, it might be helpful to use a slimmed-down version of lattice file formats. Such a restricted version would only contain basic element types such as drifts and multipoles. As these elements are available in all accelerator codes, this would make a one-to-one translation between the lattice file formats possible. Furthermore, with the lattice files available in the different formats, automated routines could be set up for the different simulations codes.

A prove-of-concept framework of automated lattice summaries was developed using the new lattice design tool and JSON-based lattice file format and will be presented in Section 4.3.

1.2.4 Smaller Lattice Development Tasks at the BESSY II Storage Ring

Sometimes there are smaller lattice development tasks, where, for example, a user needs to know the value of the beta function at a specific position in the storage ring. Often this task requires some post-processing, which at HZB is mainly done in Python. Thus, having a native Python interface to the accelerator model and the Twiss parameters would be very beneficial for these tasks.

2.3.1 Linearized Equations of Motion

To linearize Equation 2.16, we expand the expression

in \delta, leading to:

Multiplying out the parentheses and only keeping terms linear in x, y, and \delta, we obtain the linearized equations of motion:

Equation 2.20 is a linear second order differential equation, which can be solved analytically. Therefore, the transformation of the particle trajectory

through beam transport elements can be described as a matrix multiplication

where \mathbf{R} is a 6 \times 6 transfer matrix. The solutions of Equation 2.20 for a drift section (\kappa_\mathrm{x0} = k = 0), for a dipole magnet (\kappa_\mathrm{x0} \neq 0, k = 0), and for a quadrupole (\kappa_\mathrm{x0} = 0, k \neq 0) are included in Section 9.

2.3.2 Betatron Oscillation

Tracking a particle through the beam transport system by applying the different transfer matrices of beam transport elements yields the individual particle trajectory. However, Courant and Synder [19] developed a formalism, revealing more insightful analytical properties of the entire particle beam. The Courant-Snyder functions, sometimes called Twiss parameters, arise from separating the on- and off-momentum motion.

Therefore, we first solve the linearized equations of motion, Equation 2.20, for the dispersion-free case, leading us to the most fundamental quantity of the transversal beam motion, the beta function \beta(s). Then, in Section 2.3.4, we introduce the dispersion function \eta(s) to account for the transverse motions caused by momentum deviations.

By neglecting the off-momentum terms and combining the focusing terms into one parameter K_\mathrm{x}(s) = \kappa_\mathrm{x0}^2(s) + k(s) for the horizontal and K_\mathrm{y}(s) = - k(s) for the vertical plane, the linear equations of motion become

Equation 2.23, known as Hill’s equation, is a second-order linear ordinary differential equation similar to the harmonic oscillator. The difference is that the parameter K(s) = K(s + C) is not constant but is a function periodic with the circumference of the accelerator C. The Floquet’s theorem states that Hill’s equation has two linearly independent solutions, given by the product of a complex exponential function and a periodic function [20]. The real part of the general solution of Hill’s equation is

where we identify the integration constants \epsilon and \psi_0 with the emittance and the initial betatron phase, respectively. The beta function \beta_u(s) is periodic with the circumference of the accelerator C.

Figure 2.3 shows the particle trajectories and the beam envelope as defined by Equation 2.24. The individual particles oscillate within the envelope

As the beta function \beta(s) determines the shape of this envelope, and therefore the transverse beam size, it is considered one of the most important quantities in circular accelerator physics.

By substituting Equation 2.24 and its second derivative into Equation 2.23, we obtain an expression for the betatron phase

The tune

describes the number of betatron oscillations per turn.

By rearranging Equation 2.24 and its first derivative with respect to the orbit position s

we obtain an ellipse equation for the Twiss parameters

where

and

Following Equation 2.29, the trajectory of betatron oscillation forms an ellipse in the (u, u')-phase space. The Twiss parameters \alpha(s), \beta(s), and \gamma(s) determine the shape of this ellipse at the orbit position s. The emittance \epsilon, which we introduced as an integration constant, describes the occupied phase space volume of the ellipse A = \pi \epsilon, according to Liouville’s theorem a constant of motion [21].

Furthermore, supposing the Twiss parameters \alpha, \beta, and \gamma are know for the positons s and s + L. Then, the 2×2-sub-matrices \mathbf{R}_u^{2\times 2}(s,L) of \mathbf{R}(s, L) describing the transformation of a particle from the orbit position s to s + L within horizontal or vertical plane is

where \beta_0 = \beta_u(s), \beta_1 = \beta_u(s+L), \alpha_0 = \alpha_u(s), \alpha_1 = \alpha_u(s+L), and \psi_1 = \psi_u(s+L).

2.3.3 Transformation of the Twiss parameters

Equation 2.29 can be written in matrix form:

We define the Twiss matrix

Since the emittance \epsilon is the same at the position s and s + L, rearranging Equation 2.33

results in an expression for the transformation of the Twiss parameters

Multiplying the matrices of Equation 2.36 yields a system of linear equations

For a stable solution to exist, the Twiss functions must have the same value after one revolution, leading to the periodicity condition for circular accelerators

where C_0 is the length of the ideal orbit. Using this condition we can solve the system of linear equations of Equation 2.37, yielding the Twiss functions as expression of the one-turn-matrix \mathbf{R}(s, C_0):

From the expression of the beta function \beta(s) in Equation 2.39, it follows that a stable solution only exists for

defining a stability criteria.

2.3.4 Dispersion

The dispersion function

describes the change in the transverse offset u(s) with respect to the relative momentum deviation \delta. Therefore, the trajectory of an off-momentum particle is given by the sum of the betatron oscillation u_\beta(s) and the dispersive offset u_\delta = \eta_u(s) \delta:

Supposing the values of the dispersion function \eta(s) and its derivative \eta'(s) are known at the orbit position s. Then, the transfer matrix \mathbf{R}(s, L) determines their values at the orbit position s + L:

Using the periodicity condition for circular accelerators

the dispersion function \eta(s) and its derivative \eta'(s) can be expressed in terms of the one-turn-matrix \mathbf{R}(s, C_0):

To calculate the graph of \eta(s) and \eta'(s) along the entire ring, Equation 2.45 can be used to calculate the values \eta_0 = \eta(s_0) and \eta'_0 = \eta'(s_0) at an initial orbit position s_0. Afterward, Equation 2.43 yields the values of \eta(s) and \eta'(s) for all other orbit positions s.

2.3.5 Momentum Compaction

The momentum compaction factor

is defined as the ratio between the change in the path length of a dispersive particle for one revolution C with respect to the relative momentum deviation \delta and the path length of the ideal orbit C_0. While C_\beta is the path length of an on-momentum particle,

corresponds to the path length difference caused by the momentum deviation. Applying the linear approximation of the path length element \mathrm{d}z \approx (1 + \kappa_\mathrm{x0} x) \mathrm{d}s, introduced in Equation 2.12, to Equation 2.47 yields

Substituting Equation 2.48 into Equation 2.46, we obtain an expression for the momentum compaction factor

corresponding to the mean value of the product between the curvature of the ideal orbit \kappa_\mathrm{x0}(s) and the dispersion function \eta(s).

2.4.1 Natural Chromaticity in a Storage Ring

The number of betatron oscillations per turn, defined by the tune, is determined by the position and strength of the quadrupole and bending magnets. However, as the experienced multipole strength of a particle depends on its momentum, off-momentum particles will complete a different number of betatron oscillations per turn. Figure 2.4 shows the influence of the momentum deviation on the focal length of a quadrupole. Generally, particles with a higher momentum \delta > 0 are less deflected by a quadrupole, resulting in fewer betatron oscillations. On the other hand, particles with a negative momentum deviation \delta < 0 undergo a larger deflection, leading to a higher tune.

The chromaticity

describes the change in the betatron tune with respect to the relative momentum deviation \delta. In linear approximation, the perturbed quadrupole strength caused by the momentum deviation

has the same effect as a gradient error

Therefore, to derive the effect of this gradient error, we introduce a perturbation in form of a thin quadrupole

with the infinite focal lenth \frac{1}{\Delta k \mathrm{d}s}. Here we look at the 2×2-sub matrix, which only transforms the horizontal or vertical componentes of the particle vector \mathbf{X}. According to Equation 2.32 the 2×2-one-turn-matrix in the horizontal or vertical plane

can be written as an expression of the Twiss parameters, where \Psi = 2 \pi Q. Hence, we can express the perturbed one-turn-matrix in terms of the unperturbed Twiss parameters:

By calculating the trace of the perturbed one-turn-matrix

we obtain

where \Psi_\mathrm{p} = \Psi + \mathrm{d}\Delta\Psi. For small tune pertubations \Delta Q, we can use the small-angle approximiation of the trigonometric functions \cos{\mathrm{d}\Delta\Psi} \approx 1 and \sin{\mathrm{d}\Delta\Psi} \approx \mathrm{d}\Delta\Psi, resulting in an expression for the differential of the tune pertubation

By integrating Equation 2.58 and substituting it into Equation 2.50, we obtain an expression for the natural chromaticities

which only takes the contribution of the momentum-dependent quadrupole strength into account. Since particles with a positive momentum deviation (\delta > 0) experience a weaker focusing, the natural chromaticities \xi_u^\mathrm{n} are always negative.

2.4.2 Chromaticity Correction

Two effects cause chromaticities to be undesirable in circular accelerators, making it necessary to correct for these chromatic errors. First, resonances between the betatron oscillation and the magnetic fields arising at specific values of the betatron tune can lead to a beam loss. Thus, for high chromaticities, where even particles with small momentum deviations experience a large tune shift, moving particles into resonance becomes more likely. Moreover, particles have vastly different tunes for high chromaticities, corresponding to a tune spread in the tune diagram. Hence, the accelerator has to be operated at a distance to the nearest resonances larger than the tune spread, imposing strict constraints on possible tunes and making it challenging to choose an operational tune.

Secondly, so-called head-tail instabilities, a collective effect between the electrons of the head and tail of a bunch, grow proportional with the chromaticity.

Non-linear elements can compensate for the chromaticities introduced by the momentum-dependent quadrupole strength. The goal is to give particles with a positive momentum deviation \delta > 0 an additional kick and counterbalance the stronger kick, which particles with a negative momentum deviation \delta < 0 experience. Figure 2.5 shows a sextupole compensating for the chromaticity introduced by a quadrupole. A sextupole magnet behaves in one plane like an amplitude-dependent quadrupole magnet. Therefore, a sextupole has the desired effect at a position of dispersion \eta \neq 0, where the particles are sorted by their momentum deviation \delta.

Using the additional sextupole kick experienced by a dispersive particle \Delta k = m \eta \delta, we can, analog to Section 2.4.1, derive an expression for the part of the chromaticity introduced by the sextupole magnets

We obtain the total chromaticity from the sum of the natural chromaticity \xi^\mathrm{n} and the chromaticity caused by the sextupole magnets \xi^\mathrm{s}

2.5.1 Radiation Damping of Betatron Oscillations

As shown in Figure 2.6, a radially accelerated electron emits synchrotron radiation in its general direction of motion, leading to a decrease of the longitudinal and transversal components of its momentum. However, the electric field of the cavity only restores the electron’s longitudinal momentum. Over time, the interplay of both of these effects leads to a damping of the electron’s betatron oscillation. While this effect is commonly called the radiation damping, the damping of the betatron oscillations occurs in the cavity. The emission of synchrotron radiation does not - apart from slightly changing the focusing strength of the magnets due to the energy loss - directly affect the amplitude of the betatron oscillations. However, the damping is still a second-order effect of the synchrotron radiation, as, without it, the cavity would not restore the longitudinal momentum.

To quantify the influence of the synchrotron radiation on the amplitude of the betatron oscillations, we make some assumptions:

• First, we assume the emission of synchrotron radiation is precisely parallel to the momentum of the electron.
• Second, we assume a continuous emission of synchrotron radiation over a given path length.
• Finally, we assume the acceleration happens continuously along the ring.

Under these assumptions, the betatron oscillations would dampen to zero. However, due to the quantum nature of photons, the emission happens in discrete chunks of energy, making the second assumption not entirely reasonable. The discrete emissions lead to a counteracting excitation which will be considered in Section 2.5.2.

We recall the definition of the emittance

from Equation 2.29. Note that the coordinates u and u' correspond to the betatron coordinates, which do not include the offset and slope caused by the momentum dispersion. By using the above definition of the emittance, we obtain an expression for the variation of the emittance with respect to the transverse coordinates u and u'

Neither the emission of a photon nor the acceleration immediately changes the transverse displacement of an electron u, leading to \delta u = 0. The sum of the synchrotron radiation and the cavity forces results in a variation of transverse momentum

Therefore the variation of transverse slope u' is

Furthermore, we have to take into account the influence of the momentum deviation. Therefore, we will first solve the dispersion-free vertical case \eta_\mathrm{y} = 0 and then add an additional term for the horizontal case \eta_\mathrm{x}, where \neq 0.

With \delta p_\perp = -y' \delta p_\mathrm{rf} and p_\parallel \approx p, we can conclude for the variation of the vertical coordinates

and obtain an expression of the variation of the vertical emittance

where \delta E_\mathrm{rf} corresponds to the energy refilled by the cavity, not the energy decrease due to the damping \delta E = c \sqrt{p_\parallel^2 + (p_\perp + \delta p_\perp)^2} - c \sqrt{p_\parallel^2 + p_\perp^2} \approx c \frac{p_\perp \delta p_\perp}{p} \approx c y' \delta p_\perp. Under the assumption of a uniform distribution of betatron phases \psi, we must average over Equation 2.73. Therefore we solve the integrals

and

and substitute them into Equation 2.73:

Over the time of one revolution T_0, the energy chunks \delta E_\mathrm{rf} add up to the total radiation loss U_0. Therefore, following our third assumption, we write an expression for the time derivative of the vertical emittance

For the variation of the horizontal emittance

we have to consider an additional term \delta \epsilon_\mathrm{x}^{\delta} caused by the momentum dispersion. As discussed in Section 2.3.4, the total horizontal displacement x_\mathrm{tot} is sum of betatron displacement x(s) = \sqrt{\epsilon \beta(s)} \cos(\psi(s) + \psi_0) and dispersive displacement x_\delta(s) = \eta(s) \delta:

As the energy loss due to the emission of a photon cannot immediately change the total displacement x_\mathrm{tot} or slope x'_\mathrm{tot} of an electron, the betatron coordinates x and x' must compensate the change in the dispersive coordinates x_\delta and x'_\delta. Therefore the variation of the betatron coordinates x and x' is

where \delta E_\mathrm{ph} is the variation of the electron energy caused by the synchrotron radiation. Substituting Equation 2.80 into Equation 2.69 yields the additional term of the variation of the horizontal emittance

From [22], we use the expression for the radiation power

where E and B corresponds to the electric and magnetic fields, respectively, and we introduced a constant

As P \propto E^2 B^2, the radiation power is in linear approximation

Subsituting \Delta B \approx \frac{\mathrm{d}B}{\mathrm{d} x} x, \frac{\mathrm{d}B}{\mathrm{d} x} = \frac{k}{\kappa_\mathrm{x0}} B and \Delta E = E_0 \delta into Equation 2.84, yields

With t = z / c and Equation 2.85 we can write for the variation of the Energy

Using Equation 2.86, we can average over all betatron phases \psi

Only even terms in x and x' will contribute a finite value, while odd terms in x and x' vanish. Analogous to Equation 2.74 we find that \langle x x' \rangle = -\frac{\epsilon_\mathrm{x} \alpha_\mathrm{x}}{2} and for average of x^2 we obtain

which we substitute into Equation 2.86:

Using the relationships \mathrm{d} t = \mathrm{d} z / c, we define nominal radiation loss per turn

where we made use of the fact that for the nominal particle s = z. By comparing Equation 2.82 and Equation 2.90, we obtain an expression for the nominal radiation power

in terms of the curvature of the ideal orbit \kappa_\mathrm{x0} and the nominal energy loss per turn U_0. Substituting Equation 2.91 into Equation 2.89 and integrating over one turn yields the change of the horizontal emittance caused by radiation damping

where we inserted the second and fourth synchrotron radiation integrals defined in Equation 2.63 and Equation 2.65.

2.5.2 Quantum Excitation

Until now, we assumed the radiation would be emitted continuously along a given path length parallel to the particle’s momentum. If this were the case, then after some time, the betatron amplitude would damp to zero. However, due to the quantum nature of photons, the radiation is not emitted continuously but rather happens in discrete chunks of energy, resulting in an excitation of the emittance called quantum excitation. Illustrated in Figure 2.7, the discontinuous jump in energy caused by the emission of a photon of discrete energy forces the electron on a dispersive orbit p \neq p_0. Subsequently, the electron now performs betatron oscillations around the new closed orbit.

Furthermore, the photons are not emitted precisely parallel to the electron’s velocity but rather in a cone, resulting in a change in the direction of the momentum. That also excites the betatron oscillation and prevents the emittance from damping to zero, even if the emission of synchrotron radiation would be continuous. However, as the latter effect is much smaller than the former, we will neglect this effect for the following considerations.

For same argument leading to Equation 2.80, the emission of a photon with the discrete energy \delta E leads to a variation of the betatron offset and betatron slope:

Inserting \delta x and \delta x' into the definition of the phase space ellipse from Equation 2.68, we obtain an expression for the variation of the emittance caused by the emission of a discrete photon of the energy \delta E

where we substituted the curly \mathcal{H} function defined in Equation 2.67. As the photons are emitted randomly with different energies \delta E, we have to average Equation 2.94 over the energy distribution of the photons. Therefore, we introduce a new quantity \dot{n}(\epsilon) which denotes number of photons emitted per time unit in the energy interval [\epsilon, \epsilon + \mathrm{d} \epsilon].

According to [22], the squared emitted energy per time unit is

where

is to the so-called quantum constant.

Using Equation 2.95, we can integrate Equation 2.94 along the length of the ring to obtain an expression for change of the horizontal emittance caused by the quantum excitation

where in the last step, we substituted the second and fifth synchrotron radiation integrals I_2 and I_5 defined in Equation 2.63 and Equation 2.66. Note that the change of the emittance is positive, which corresponds to an excitation of the emittance.

2.5.3 Equilibrium Emittance

As discussed in the last two subsections, two counteracting effects are changing the amplitude of the betatron oscillations. The total time derivative of the emittance corresponds to the sum of both, i.e., Equation 2.92 and Equation 2.97:

While the synchrotron damping leads to an exponential decay of the emittance, the quantum excitation is independent of the amplitude causing the emittance to grow constantly. Therefore eventually, both effects become equally strong, leading to a zero time derivative of the emittance

at a value known as equilibrium emittance

Note that the magnetic lattice fully defines Equation 2.100. For electron storage rings, this has a very practical consequence: The equilibrium emittance is unrelated and therefore not limited by the emittance of the source. Consequently, any arbitrary injected particle distribution damps to the value of the equilibrium emittance.

In the literature, the expression for the equilibrium emittance often differs by a factor of two from Equation 2.100. The reason is that many books use the root-mean-square of the horizontal betatron displacement

to establish an alternative definition of the emittance

2.6.1 The FODO Lattice

The FODO cell is the simplest possible strong-focusing lattice. It consists out of alternating horizontal and vertical focusing quadrupoles with drift spaces in between, which give the cell its name: A horizontal Focusing quadrupole, a drift space (0 force), a horizontal Defocusing, and a drift space (0 force). A schematic of the FODO lattice is shown in Figure 2.8.

We can use the periodic condition of Section 2.3.3 to calculate the optical functions \beta(s) and \eta(s). The periodic solution of the Twiss parameters \beta(s) and \eta(s) for a FODO lattice without dipoles (R = 0) are shown in the upper plot of Figure 2.9.

While the horizontal beta function \beta_\mathrm{x}(s) reaches its maximum at the center of the horizontal focusing quadrupole q1, the vertical beta function \beta_\mathrm{y}(s) has its maximum at the center of the vertical focusing quadrupole q2. As the dispersion is introduced by bending magnets, this FODO cell has a vanishing dispersion function \eta_\mathrm{x}(s). A high dispersion function is especially undesirable at the location of insertion devices, which would significantly increase the quantum excitation. A circular accelerator must have bending magnets, as - by definition - it has to close at some point, which makes a non-vanishing dispersion function \eta_\mathrm{x}(s) inevitable. However, as discussed in the following subsection, it is still possible to design a lattice with no dispersion within certain sections of the ring.

The simplest way to create a FODO-based circular accelerator is to place a bending magnet in the center of every drift space. The Twiss parameters \beta(s) and \eta(s) for such a FODO cell with dipoles (R = \frac{\pi}{8}) are shown in the lower plot of Figure 2.9. Table 2.1 lists a comparison of the lattice parameters for both FODO cells.

As one can see, the dipoles introduced a non-zero dispersion function \eta_\mathrm{x}(s) with a graph similar to the horizontal beta function: A maximum of 1.8 meters at the center of the horizontal focusing quadrupole q1 and a minimum of 0.8 meters at the center of the vertical focusing quadrupole q2. Due to the horizontal weak-focusing of the dipoles and vertical focusing of the dipole edges, the maxima of the horizontal and vertical beta functions are slightly smaller than in the FODO cell without dipoles. For the same reasons, the vertical tune Q_\mathrm{y} is a bit larger.

Discussed in Section 2.3.2, the evolution of phase space ellipse of the beam is fully defined by the \beta_u(s), \alpha_u(s) and \gamma_u(s) functions. Figure 2.10 shows the graph of the \beta_u(s) and \alpha_u(s) functions together with betatron phases \psi_u(s). According to their definition \alpha_u(s) = -\beta_u'(s) / 2 , the alpha functions \alpha_u(s) become zero when the beta functions \beta_u(s) are maximum and reach their maximum when the beta functions \beta_u(s) change the most. The horizontal betatron phase \psi_\mathrm{x}(s) changes the most in the vertical focusing quadrupole q2 and its adjacent drift spaces. In contrast, the changes within the starting and ending q1 quadrupoles are almost negligible. For the vertical betatron phase \psi_\mathrm{y}(s), it is the other way around. However, due to the edge focusing of the bending magnets, the total vertical phase advance Q_\mathrm{y} is slightly larger than in the horizontal plane.

The product of the periodic beam amplitude \sqrt{\epsilon\beta(s)} and the cosine of the betatron phase \psi_u(s) defines the trajectory of a particle in a strictly linear lattice

as shown at the example of four consecutive FODO cells in the upper part of Figure 2.11. The particle performs betatron oscillations according to its betatron phase \psi_u(s) + \psi_{u,0} within the beam envelope \sqrt{\epsilon\beta(s)}. The second plot of Figure 2.11 shows the same trajectory using Floquet’s coordinates u(s) / \sqrt{\beta(s)}. In these coordinates, the beam envelope \sqrt{\epsilon\beta(s)} becomes a constant and the particle performs oscillations distorted by the betatron phase \psi_u(s). When plotting the particle trajectory against the betatron phase \psi_u(s) instead of the orbit position s, shown in the third plot of Figure 2.11, the oscillation has a perfect sinusoidal shape. Note that in this representation, the elements of magnetic lattice become distorted according to their phase advance \psi_u(s_1,s_2).

Figure 2.12 shows a scan over possible quadrupole settings for FODO cells with different lengths L and deflection angles, where we used the stability condition from Equation 2.40. Not all configurations of quadrupole strengths result in a stable lattice. One condition is that the focal length of a quadrupole must be greater than its distance to the next quadrupole. Assuming the focal length would be smaller, then the transverse displacement u of a particle and its derivative \frac{du}{ds} would have the same sign at the start of the following quadrupole, leading to a negative feedback loop. For each pass-through, the deflection would become stronger and stronger in the next quadrupole, eventually leading to the loss of the particle beam.

The area of stable solutions for a FODO cell without bending magnets is symmetrical to the diagonal. For low quadrupole strengths, both quadrupoles need to have a similar strength to create a stable lattice. With increasing quadrupole strength, a larger deviation of both quadrupole strengths is possible, leading to the stable area’s necktie shape. The maximum quadrupole strength is limited by the condition that the focal length of the quadrupoles must be greater than their distance from each other. The smallest average beta function \beta_\mathrm{mean} for a FODO cell with a length of 6 meters without bending magnets is achieved when both quadrupoles are set to about 1.1 m^{-2}.

With increasing cell length the area of stable solutions decreases. This is again due to the requirement that the focal length of the quadrupoles must be larger than their distance. This means that moving the quadrupoles closer together allows for greater field strengths.

The second and third row of Figure 2.12 shows the stability plots for different FODO cells with dipoles with bending angle per cell of \pi / 8 and \pi / 4, respectively. The dipoles edges lead to an additional defocusing in the horizontal plane and focusing in the vertical plane. Therefore, the necktie plot is shifted towards smaller values on the axis of the vertical focusing quadrupole q2 depending on the dipole’s deflection angle. Furthermore, dipole edge’s vertical focusing widens the space of stable configurations for lower quadrupole strengths, as a stable solution is possible even if the quadrupole q2 is turned off.

The smallest average beta function \beta_\mathrm{mean} for a FODO cell with a deflection angle of \pi / 8 per cell and with a length of 6 meters is achieved when the horizontal focusing quadrupole is set to 0.8 m^{-2} and the vertical focusing quadrupole is set to 1.1 m^{-2}. Similar to the FODO cell without dipoles, the area of stable solutions is increases when the cell length decreases.

The left part of Figure 2.13 shows the influence of the quadrupole strength on the equilibrium emittance of a FODO cell, where both the horizontal and vertical focusing quadrupoles have the same strength. As one can see, due to the choice of stronger quadrupole, it is possible to reduce the emittance by multiple orders of magnitude. Unfortunately, as shown in the right part of Figure 2.13, the chromaticity rises with stronger quadrupole values, limiting the achievable emittance as the compensation of the chromaticities leads to a limitation of the dynamic aperture.

All in all, the FODO lattice is less optimal for modern low-emittance synchrotron light sources. For high-energy colliders, the FODO lattice is useful as it maximizes the ratio of bendings magnets per circumference. However, in a synchrotron facility, one generally wants to maximize the ratio of free straights per circumference to utilize insertion devices. Furthermore, the high dispersion function of the FODO lattice leads to a large equilibrium emittance, which is increased even more due to the additional quantum excitation of insertion devices at the location of high dispersion functions. Therefore, the following section presents a lattice with a smaller equilibrium emittance that is more suitable for insertion devices.

2.6.2 The DBA Lattice

As discussed in Section 2.5.2, quantum excitation increases the emittance of the particle beam, which is a crucial quantity for synchrotron radiation users. The double bend achromat (DBA) lattice, also known as Chasman–Green lattice [24], is one type of lattice to achieve a lower emittance than with the FODO lattice presented in the last subsection. In addition, the DBA lattice’s dispersion-free sections are suitable for placing insertion devices, minimizing their contribution to the quantum excitation.

Figure 2.14 shows the working principle of a DBA cell: In its simplest form, the DBA consists of two dipoles and one quadrupole. The horizontal focusing quadrupole stands between two bending magnets. The quadruple strength is chosen in such a way that it reverses the gradient of the dispersion function \eta_\mathrm{x}'(s) so that the second bending magnet cancels out the dispersion \eta_\mathrm{x}(s) introduced by the first dipole.

Not only the quadrupole’s strength but also its distance to the surrounding dipoles must be chosen correctly to fulfill the achromatic condition. Figure 2.15 shows the dispersion function \eta_\mathrm{x}, its derivative with respect to the orbit position \eta_\mathrm{x}' and the trajectory of multiple dispersive particles for three different DBA cells, where the quadrupole strengths k are identical, but the distance between bending magnets and quadrupole is different. The color of the particle trajectories indicates if the momentum deviation is positive or negative. In analogy with the rainbow spectrum of visible light, a particle trajectory drawn in color towards purple represents a positive momentum deviation \delta > 0. In contrast, a color towards red corresponds to a negative momentum deviation \delta < 0.

In the upper plot, the distance between the dipole magnets and the quadrupole is too short. Here, the dispersion function \eta_\mathrm{x} did not have enough space to build up so that the quadrupole could fully reverse the gradient of the dispersion function \eta_\mathrm{x}'. Consequently, after the beam exits the cell, there is a non-vanishing dispersion function \eta_\mathrm{x} > 0. Alternatively, one could increase the quadrupole strength k to meet the achromatic condition. In the second case, the quadrupole strength k and the distance between the dipole magnets and the quadrupole match perfectly. As a result, the quadrupole exactly reverses the gradient of the dispersion function \eta_\mathrm{x}' so that the dispersion \eta_\mathrm{x} outside the cell vanishes. In the last plot, the distance between the bending magnets and the quadrupole is too large, resulting in a negative dispersion function \eta_\mathrm{x} < 0 at the end of the cell. In this case, choosing a lower quadrupole strength k would satisfy the achromatic condition.

Usually, there are additional quadrupoles outside the achromat to form a complete DBA cell. Figure 2.16 shows the graphs of the beta functions \beta_u(s), the dispersion functions \eta_\mathrm{x}(s) and \eta_\mathrm{x}'(s) and the betatron phases \psi_u(s) of a DBA cell with two additional quadrupoles q1 and q2 outside of the achromat. Additionally, Table 2.2 lists important lattice parameters of this DBA cell.

Here the quadrupoles q1 and q2 provide the horizontal and vertical focusing of the particle beam. As discussed above, the quadrupole q3 reverses the gradient of the dispersion function \eta_\mathrm{x}'. For the second dipole to have precisely the opposite effect on dispersion as the first, the particles must perform exactly half a betatron oscillation from the center of the first dipole to the center of the second dipole. This condition corresponds to a phase advance of \psi_\mathrm{x}(s_\mathrm{center_1}, s_\mathrm{center_2}) = \pi, marked in the bottom plot of Figure 2.16.

Not all quadrupole configurations satisfy the stability condition from Equation 2.40, making it significantly more difficult to find the optimal quadrupole setting for a given objective. In the case of the FODO cell, which has only two parameters, it is relatively straightforward as it is possible to scan over all possible quadrupole settings, as shown in Figure 2.12. Furthermore, even though unstable lattice configurations exist, there is only one contiguous area of stable configurations. However, as shown in Figure 2.17, even for the simple DBA lattice of Figure 2.16, which only has three different quadrupole families, there emerge multiple areas where a stable solution exists. While it is for three quadrupole families still possible to scan over all possible values, this becomes unfeasible for six or more magnets [5]. These discontinuities, leading to the islands of stable lattice configurations, make optimizers only useable within a given island and therefore pose one of the most challenging tasks in lattice development. Section 4.1.3 discusses how some of these problems have been dealt with for the Q5T2off lattice of the BESSY II storage ring.

Overall, the DBA lattice is suitable for a low emittance synchrotron light source and therefore was a very common choice among many third-generation light sources, including BESSY II. However, the multi-bend achromat (MBA) lattice allows for even smaller emittances, making it a popular choice for fourth-generation light sources. The additional quadrupoles placed between the bending magnets keep the dispersion function further down, resulting in a smaller emittance.

Note that the achromatic condition does not necessarily lead to the smallest equilibrium emittance, as defined in Equation 2.100. As demonstrated in [25], another trend is to break the achromatic condition and leak a finite value of the dispersion function \eta_\mathrm{x} into the straight sections. That effectively distributes the dispersion function \eta_\mathrm{x} along the ring and reduces it within the dipole magnets. As long as the dispersion function \eta_\mathrm{x} does not become too large within the insertion devices, distributing the dispersion function over the entire ring can reduce the equilibrium emittance.

4.1.1 Lattice Design of the BESSY Storage Ring

The BESSY II storage ring has a double bend achromat lattice with 16 straight sections. The injection requires high horizontal beta functions. At the same time, the superconducting wavelength shifter needs a small horizontal beta function. That led to the decision to develop a lattice with alternating high and low horizontal beta straights [43]. As a result, the 240 m storage ring has an 8-fold symmetry with the unit cell shown in Figure 4.1. The seven quadrupole families of the design-lattice are listed in Table 4.1.

The two horizontal focusing Q1 quadrupoles in the center of the DBA reverse the gradient of the dispersion function as discussed in Section 2.6.2. The two vertical focusing Q2 quadrupoles are necessary to limit the vertical beam size. The high beta straights have the quadrupole doublet Q3D and Q4D, which are vertical and horizontal focusing, respectively. The low beta straights have the vertical focusing Q3T, horizontal focusing Q4T, and vertical focusing Q5T quadrupole families, which form a triplet. In order to achieve the low horizontal beta function, the Q4T quadrupoles must be significantly stronger than the Q4D quadrupoles. The additional Q5T quadruples are needed to compensate for the stronger vertical defocusing introduced by the Q4T quadrupoles.

Turning off the Q5T2 quadrupoles means turning the T2 triplet section into another doublet section. As the BESSY II storage ring has undergone several modifications since the design lattices from 1996, there are additional constraints that have to be taken into account:

Since 2005, a femto-slicing facility, producing ultra-short x-ray pulses, is commissioned in the D6 section of the BESSY II storage ring [44], [45]. A femtosecond laser pulse modulates the electron energy in a wiggler called the modulator. The off-momentum electrons are extracted by the transverse displacement of a bending magnet, and the synchrotron radiation is emitted in the following undulator called the radiator. This femto-slicing experiment required the installation of three additional bending magnets, which together form a dipole chicane, the wiggler U139, and the undulator UE56. Another significant alteration of the BESSY II storage ring was carried out as part of the EMIL project [46], which is dedicated to researching materials for renewable energy. It provides multiple beamlines with simultaneous access to soft and hard x-rays. These are provided by the two UE-48 and CPMU-17 undulators, which are canted to separate the generated beam cones. In order to support this setup of the two canted undulators, the vertical focusing quadrupole QIT6 was installed in the center of the T6 straight, which shifts the vertical beam waist to the center of the CPMU-17 device. The Twiss parameters in D6 (Emil) and T6 (femto-slicing) straights are shown in Figure 4.2.

Another modification of the BESSY II storage ring was the introduction of the so-called injection optics [47]: The horizontal beta function \beta_x was increased in the injection-straight D1 and decreased in all other doublet straights to improve the injection efficiency.

4.1.2 Constraints for the Development of the Q5T2off-optics

The Twiss parameters of the BESSY II storage ring were carefully tuned to support the mentioned changes. New modifications should not conflict with the lattice development of the last several years. Therefore, when turning off the Q5T2 quadrupoles, it should be made sure that the changes are as local as possible with respect to the T2 section, and perturbations of the Twiss parameters outside of the T2 should be as small as possible.

The VSR cavities set another requirement for the Q5T2off-optics. As addressed by [1] beam-cavity interactions can cause coupled bunch instabilities, which spoil the beam quality. For a fixed threshold transverse cavity impedance

the average beam current I_{\mathrm{DC}} is determined by the energy E, the damping rate \tau_{\mathrm{d}}^{-1}, the angular beam revolution frequency \omega_\mathrm{rev} and the value of the beta function \beta within the cavity. As the requirements of the users of the synchrotron radiation restrain the energy E, the circumference of the ring defines the angular beam revolution frequency \omega_\mathrm{rev} and increasing the damping rate \tau_{\mathrm{d}}^{-1} might conflict with aspects of the machine operation, the value of the beta function within the cavity module effectively limits the average beam current I_{\mathrm{DC}}. By the estimations of [1, p. 83], a beta function of below 4 m within the cavity module would be sufficient to store the required current.

Assuming a symmetry point, the beta function \beta(s) at the distance from the center of T2 is fully defined by the minimum beta function \beta_{min} at symmetry point:

Therefore it is possible to calculate the required minimum beta function \beta_{min} to keep the beta function belows 4 m within the cavity moduel. Figure 4.4 shows the maximum and average beta function at the 1.50 GHz and 1.75 GHz cavities as a function of the minimum beta function \beta_{\mathrm{min}}. A minimum beta function \beta_{\mathrm{min}} between 0.6 m and 3.4 m is sufficient to keep the average beta below 4 m. A minimum beta function \beta_{\mathrm{min}} between 1 m and 2 m would be optimal to keep the average beta function within the cavity module as small as possible.

4.1.3 Turning off the Q5T2 Quadrupoles

As stated above, the goal is to minimize the beta functions’ perturbation compared to the current standard user optics and keep the changes as local as possible. This can be accomplished by an optimizer that minimizes a fitness function chosen to satisfy the stated objective. However, the fact that not all quadruple settings result in a stable lattice poses a unique challenge: As an unstable quadrupole setting has no solution, all of them are equally bad. Therefore, choosing an objective function that provides a practical value to the optimizer in such a case is not easy. It seems reasonable to return a high constant or infinity. However, this leads to a problem when starting an optimizer in a region of unstable quadrupole settings. Usually, all other quadrupole settings in configuration space in the direct vicinity of an unstable lattice configuration are unstable as well. Therefore, the objective function always returns the same value regardless of which direction the optimizer chooses in the configuration space. Thus, the optimizer is practically left in the dark, and the possibility to converge is left to chance.

Just switching off the Q5T2 quadrupoles in the current standard user optics leads to an unstable lattice. That means that a quadrupole setting with only the Q5T2 quadrupoles turned off cannot be used as a starting configuration for optimization. So first, a stable lattice configuration has to be found, which can then be further improved by an optimizer.

A reasonable approach seems to start by only using quadrupoles in the T2 section to find a stable quadrupole setting. This was directly tested at the machine and also afterward checked by doing a quadrupole scan in simulations. The stability condition of the Twiss parameters

was derived in Section 2.3.3. The quadrupole scan in Figure 4.5 shows the stable regions for different settings of the Q3T2, Q4T2, and Q5T2 magnets.

At the machine, the first idea was to use the other vertical focusing Q3T2 quadrupoles to compensate for the turn-off of the Q5T2 quadrupoles. Increasing the Q3T2 quadrupoles made it possible to slightly more decrease the Q5T2 quadrupoles. However, at about 94 % of the initial strength of the Q5T2 quadrupoles, the beam was lost. That seems to be consistent with the results of the quadrupole scan. The blue cross in Figure 4.5 marks this unstable quadrupole setting.

The next idea was to compensate for the turn-off of the Q5T2 quadrupoles by reducing the strength of the vertical defocusing Q4T2 quadrupoles. That led to a working machine but resulted in a very low injection efficiency. Figure 4.6 shows the Twiss parameters where only the Q3T2 and Q4T2 quadrupoles were changed to compensate for the turn-off of the Q5T2 magnets. Compared to the standard user optics, the perturbation of the beta functions is significant all around the ring, which probably led to the low injection efficiency. In particular, the vertical beta function increases substantially in the T1, T3, and T6 sections.

The first and most local solution provides a starting point to improve the optics further using an optimizer. The average relative change of the beta function

was chosen as the objective function. Different combinations of quadrupoles were used to compensate for the turn-off of the Q5T2 quadrupoles. Motivation for selecting the different quadrupole sets and a detailed overview of the solutions can be found in my bachelor thesis [5]. Figure 4.7 shows the best solution obtained within the scope of the bachelor thesis. The optics was tested at the storage ring, and a high current with reasonable lifetime and injection efficiency was stored.

4.1.4 Optimizing the Q5T2off-optics in Simulations

As stated in the conclusion of my bachelor’s thesis, further improvements to the optics are necessary: First, the presented solution has a non-negligible \beta-beat outside of the T2 section. The standard user optics is fine-tuned in the injection straight, EMIL and, femto-slicing sections. Therefore, changes introduced by the Q5T2off optics should be as local as possible to avoid conflicting with previously made considerations. The goal is to limit the \beta-beat to the T2 and its adjacent neighboring sections. Another issue is that the vertical beta function in the center of the T2 section is with 3.6 m still too high. As stated before, the minimum beta function must be a least 3.4 m and would optimally be between one and two meters. Furthermore, the optics have to be further optimized regarding the non-linear dynamics. Different sextupole settings can be used to optimize the momentum acceptance.

Adjusting the objective function can reduce the \beta-beat. My bachelor’s thesis used the mean relative residual of the beta function. That meant that a change from 4 m to 2 m corresponding to -50 % had the same weight as a change from 20 m to 30 m (+50 %). Therefore, this incentivized the optimizer to reduce the beta function below the value of the standard user optics at certain positions, resulting in a larger \beta-beat. This can be solved by composing the relative change of the beta function \frac{\beta(s)}{\beta_\mathrm{ref}(s)} with a rectified linear unit (ReLU) function R(x) = \mathrm{max}(1,x), which maps every value below one to one. This has the effect of a lower threshold, which disincentivizes to lower the beta function below the reference value at the expense of making it larger somewhere else. Larger changes can be more disincentivized by squaring the relative change. To ensure the vertical beta function \beta_\mathrm{y} is small enough at the center of the T2 section s_\mathrm{T2}, it can be included in the objective function.

Taking the mentioned considerations into account, leads to the new objective function:

With the developed code, this roughly translates to:

def objective_function(values, quads):
    for quad, value in zip(quads, values):
        quad.k1 = value

    if not twiss.stable:
        return np.inf

    beta_beat_x = np.maximum(1, twiss.beta_x / twiss_ref.beta_x) ** 2
    beta_beat_y = np.maximum(1, twiss.beta_y / twiss_ref.beta_y) ** 2
    return np.mean([beta_beat_x, beta_beat_y]) + twiss.beta_y[t2_center]

The optimization procedure was run for different combinations of quadrupoles. Due to the improvements made to the Twiss calculation code, it was now possible to do much more iterations in a much shorter time. Furthermore, it was possible to run the optimization procedure multiple times in a row, using the last optimization result as a starting point for the next. Figure 4.8 shows the Twiss parameters of the best obtained Q5T2off optics compared to the current standard user optics. A complete code snippet that reproduces the Q5T2off optics and explanation is included in the Section 7.

Figure 4.9 shows the \beta-beat of the new Q5T2off optics compared to the old Q5T2off optics. The old Q5T2off optics has a high horizontal \beta_\mathrm{x}-beat in the injection straight and a high vertical \beta_\mathrm{y}-beat all around the ring, including the femto-slicing section D6 and EMIL section T6. With the improvements to the Q5T2off optics, the \beta-beat outside of T2 and its adjacent sections is effectively negligible. For the horizontal plane, the changes within the T2 and its adjacent sections are primarily negative. For the vertical plane, there are two peaks at the beginning of the adjacent DBA. Furthermore, the vertical beta function at the center of the T2 section was reduced from 3.6 m to 1.8 m.

4.1.5 User Operation Acceptance Test of the Q5T2 Optics

With improvements made to the Q5T2off optics, the following steps are to transfer the optics to the machine and test if it is ready for standard user operation. According to [18], the quadrupole strength is proportional to the current I

where \mu_0 is the vacuum permeability, a is the aperture radius and n corresponds to the winding number. Therefore, the new power supply values can be calculated by the ratio of the new and old quadrupole strength times the old power supply value:

To make sure that the optics is transferred correctly to the machine, a precise measurement of the current quadrupole strengths k_\mathrm{old} is necessary. Therefore, the Linear Optics From Closed Orbits (LOCO) method [48], [49] from the MatLab Middle Layer [50] can be used to determine the linear optics. The LOCO method first measures the orbit response matrix and the dispersion function. Afterward, the data is fitted to a lattice model to calculate the individual quadrupole strengths.

LOCO-measuring the current standard user optics started of the machine commissioning. Then a new Q5T2off optics was fitted based on the just LOCO-measured standard user optics. Next, the new power supply values were transferred to the machine using a program developed for this purpose, shown in Section 8. After the machine was set to the new Q5T2off optics, another LOCO measurement was carried out to ensure the optics was transferred to the machine correctly. Figure 4.10 shows the Twiss parameters of the LOCO measured Q5T2off optics compared to the Twiss parameters of the Q5T2off optics obtained from the optimization. There seem to be some deviations in the horizontal beta function. Especially within some of the doublet sections, the beta function is slightly asymmetric to the center. However, the vertical beta function matches very closely with the one from the simulation. Overall, it can be said that the optics was transferred correctly. Finally, the quadrupole setting of the Q5T2off optics was saved in the control software to make it available for future machine commissions. Figure 4.11 shows the LOCO-measured Q5T2off optics compared to the current standard user optics.

In another machine commissioning session, the Q5T2off optics was audited for standard user operation. During this user acceptance test, several smaller checks were performed. First, a small injection with low current was done to verify that the optics was correctly restored from the control software. Then, the quadrupoles were cycled to eliminate possible hysteresis effects. The orbit correction was used to improve the orbit. Next, a high current test was carried out. The current was injected up to 250 mA with injection efficiencies between 90% to 95%.

Figure 4.12 shows the lifetime measurement. A vertical noise is used to inflate the vertical beam size, which reduces the probability of electron collisions within a bunch and therefore increases the lifetime. Touschek lifetime \tau_\mathrm{T} and gas lifetime \tau_\mathrm{G} are 8.5 and 21.4 hours, respectively. All in all, the measured lifetimes are close to the lifetimes of standard user optics.

Up next was the measurement of the kicker lifetimes. Here the kicker magnets of the orbit bump are fired rapidly without actually injecting a beam. If the lifetime is decreased during this process, electrons are lost at the septum, which would limit the possible injection efficiency. However, no effect on lifetime could be detected, which indicates an infinite kicker lifetime.

Table 4.2 list the results of the chromaticity measurement, containing the values of the tune, first and second-order chromaticity for the Q5T2off optics compared to the standard user optics. The horizontal, vertical, and longitudinal tunes are unchanged. However, the vertical and longitudinal chromaticities of the Q5T2off optics are slightly smaller than in the standard user optics. Also, the 2ⁿᵈ order chromaticities are slightly smaller in the horizontal and vertical plane.

The streak camera was used to measure the length of the different bunches of the BESSY II fill pattern. Single bunch, PPRE bunch, slicing bunch, and multi-bunches had the same length as in the standard user optics. That was expected as the Q5T2off optics did not change the momentum compaction factor \alpha_\mathrm{c}.

As demonstrated in [47], improving the phase acceptance can increase the momentum acceptance, injection efficiency, and Touschek lifetime. At BESSY II, the longitudinal phase between the booster synchrotron and the storage ring can be varied. This allows measuring the injection efficiency as a function of the phase offset, also called a phase acceptance scan. The resulting curve’s full width at half maximum (FWHM) can then define the phase acceptance. Figure 4.13 shows the phase acceptance of Q5T2off optics compared to the standard user optics. Surprisingly, even with non-optimized sextuples, the phase acceptance seems wider but slightly lower than in the standard user optics. The FWHM of the curves are 1.2 ns and 0.9 ns, respectively. The fact that the phase acceptance of the standard optics was so small leaves the suspicion that the sextupole setting was not fully optimized during machine commissioning. Also demonstrated in [47], a phase acceptance of 1.5 ns FWHM for standard user optics is possible with an optimized sextupole setting.

The optics was run overnight as a long-term test. One thing that remains is to optimize the harmonic sextupole setting. However, all in all, the user acceptance test can be considered a success. Important beam parameters like lifetime, injection efficiency, phase acceptance, chromaticity, and kicker lifetime are as good as in the standard user optics.

4.3.1 Architecture

The source code of the lattice-summaries framework can be found on GitHub under the nobeam organization (https://github.com/nobeam).

The framework is split up into three different repositories:

1. lattice-summaries-data: A database of lattices files and optional run parameters
2. lattice-summaries: A set of routines leveraging different simulation codes
3. lattice-summaries-website: A website to inspect the simulation results

Figure 4.16 gives a schematic overview of the lattice summaries architecture. A user uploads a new lattice file and optional run files to the lattice-summaries-data repository, which functions as the database. Then routines defined in the lattice-summaries repository are run for all selected lattices. Next, the simulation results are uploaded and served by a static web server. Now the simulation results can be viewed through a website, which dynamically creates the summary views for the given simulation data. The lattice-summaries-website repository contains the source code of the website.

The presented architecture was chosen to offer maximum flexibility: Keeping the data repository and repository containing the simulation routines separate makes it possible to support multiple instances. This way, the data repository can be switched out so that different facilities can create their own instances for private lattice development. A public instance, for example, could be used to create an updated version of the Synchrotron Light Source Data Book. Furthermore, this allows the framework to be used for local lattice development, as a local instance of the data repository can be used.

The repository containing the website and the simulations results are also kept separate. That has the advantage that the website does not have to be updated when new simulation results are computed. That is convenient in the case of local lattice development, where the simulation results change frequently.

4.3.2 Database of Lattice Files

As most lattice file formats are plain text files, using Git seemed reasonable to build the database. The alternative would have been to use some NoSQL database and built a small API on top of it so that people could contribute and review new lattices. However, this seemed like unnecessary work as most of the functionality is already provided by the Git ecosystem. For example, hosting systems for Git repositories like GitHub or GitLab already provide the functionality to contribute, review contributions and run custom code in the event of a new contribution. Furthermore, Git has the advantage that it is ubiquitous, and many physicists are already familiar with how to use it.

The lattices are split into different namespaces so that multiple physicists can work on similar lattices without creating a name collision. In general, the lattice files follow the naming schema:

<namespace> / <machine>_<familiy>_v_<version>

An exemplary folder structure of a lattice database looks like:

database
├── namespace-1
│  ├── bessy2_design-1996_v_1.json
│  ├── bessy2_stduser-2019-05-07-v_1.json
│  └── info.toml
├── namespace-2
│  ├── bessy3_5ba-20p_v_long-bend-tgrb.lte
│  ├── bessy3_5ba-20p_v_reference.lte
│  └── info.toml
...

Every namespace contains an info.toml file, which contains additional metadata. That includes the human-readable title, an optional description, a list of authors, a list of labels, and a list of simulation routines that should be run for the given lattice. The list of simulation routines is necessary because a lattice file might contain special elements unique to a given simulation, not available for the other simulation routines. Possible labels, for example, could be if the lattice contains anti-bends, combined functions magnets, or longitudinal gradient bends. A detailed explanation of adding a lattice to the database can be found in the README.md file of the lattice-summaries-data repository.

4.3.3 Set of Routines

The set of routines are responsible for generating the simulation results. This is arguably the most complex part. These routines can be dependent on each other. For example, to run the routine that plots the Twiss parameters, a routine that calculates them must be run first. If the lattice is not available in the necessary format, another routine has to be run before translating the lattice file to the needed format.

The goal is that different simulation codes can power these routines, with more routines being added over time. Currently, the lattice-summaries framework provides the following routines: One routine that translates lattice files between the LatticeJSON format and the lattice file formats used by elegant and MAD-X. One routine extracts general information about the lattice, such as the circumference, energy, number of sections, section length, or bends per section. Three routines that compute the Twiss parameters using either elegant, MAD-X, or the developed code for this thesis. Furthermore, three routines that plot the Twiss parameters, a floor plan, or the higher-order chromaticity.

For local lattice development, the users change a config file to use their private version of the lattice database if they need a lattice not available in the shared database. Another convenience feature is that the dependency relation between the routines is lazy evaluated, which means that simulation is not rerun if a user only adjusts code, which is part of a visualization routine. Detailed instructions on the setup and usage are provided in README.md file of the lattice-summaries repository.

4.3.4 Lattice Summaries Website

This website provides a frontend to view the simulation results. The website is developed as a Single Page Application (SPA) using the JavaScript Framework Vue.js 3 [51] and the build tool Vite [52]. A SPA is a website that, on user interaction, dynamically rewrites the current web page instead of fetching a pre-rendered page from a server. The advantage is that the website does not have to be updated if new simulation results are uploaded. Instead, new pages are created dynamically in the frontend. Furthermore, it makes it possible to use the website for local development. By changing the DATA_URL environment variable, the user can display local simulation results.

Currently, the website consists out of three views:

1. A landing page that lists all lattices
2. A lattice view, which lists an overview of the available summaries for that lattice
3. Different types of summary views depending on the simulation routine

Figure 4.17 shows a screenshot of the landing page. It provides a search bar where lattices can be filtered by name. Furthermore, lattices can be filtered by Namespace, Machine, and Author. The search results are displayed as cards. These cards provide a short description, links to the corresponding lattice file in different formats, and links to the different types of summaries available for the given lattice.

Figure 4.18 shows an exemplary lattice summary of the BESSY II design lattice generated by the apace simulation code. The lattice summary consists out of different cards which provide different information. Independent of the type of summary, the first card is an info-card identical to the card shown in the search results. All other cards are specific to the type of summary. For example, in the case of the Twiss summary generated by the apace code, the second card provides some general information on the lattice. That includes energy, circumference, number of bends, number of sections or, section length. The third card provides a plot with the beta function and horizontal dispersion function for the unit cell. Other lattice parameters like the momentum compaction factor, emittance, or the synchrotron radiation integrals are listed in a table on the fourth card. The last card shows a floor plan of the unit cells.