Analytical considerations for linear and nonlinear optimization of the TME cells. Application to the CLIC pre-damping rings

The theoretical minimum emittance cells are the optimal configurations for achieving the absolute minimum emittance, if specific optics constraints are satisfied at the middle of the cell's dipole. Linear lattice design options based on an analytical approach for the theoretical minimum emittance cells are presented in this paper. In particular the parametrization of the quadrupole strengths and optics functions with respect to the emittance and drift lengths is derived. A multi-parametric space can be then created with all the cell parameters, from which one can chose any of them to be optimized. An application of this approach are finally presented for the linear and non-linear optimization of the CLIC Pre-damping rings.


I. INTRODUCTION
High brilliance or luminosity requirements, for electron storage or linear colliders' damping rings, necessitate ultra low emittance beams. Under the influence of synchrotron radiation, the theoretical minimum emittance (TME) [1], is reached for specific optics conditions, including a unique high cell phase advance [2]. The strong focusing needed for accomplishing the TME conditions results in cells with intrinsically high chromaticity. The chromatic sextupoles' strengths are enhanced by the low dispersion of the TME cell and reduce the Dynamic Aperture (DA). The ultimate target of a low emittance cell designer is to build a compact ring, attaining a sufficiently low emittance, with an adequately large DA, driven by geometrical aperture and injection requirements. The lattice design, however, is often based on numerical tools whose optimization algorithms depend heavily on the initial conditions. Reaching the optimal solution necessitates several iterations, without necessarily having a global understanding of the interdependence between a series of optics parameters and knobs. Modern techniques, as the Multi-Objective Genetic Algorithms (MOGA) [3] or the Global Analysis of Stable Solutions (GLASS) [4] attempt to achieve a global optics optimization exploring numerically all possible solutions, within stability and performance requirements. In this paper, a different approach is followed, by obtaining an analytical solution for the quadrupole strengths and a complete parametrization of the TME cell, using thin lens approximation. In this way, all cell properties are globally determined and the optimization procedure following any design requirement can be performed in a systematic way.
Although approximate, the obtained solutions are very close to the real thick-element optics and can be used as initial conditions for efficiently matching the lattice through numerical optics codes.
The CLIC pre-damping rings offer an ideal test-bed for applying the procedure mentioned above: they have to accommodate a large emittance beam, coming in particular from the positron source and reduce its size to low enough values for injection into the main damping rings. The latter requirement imposes a low emittance cell linear optics design, whereas the former one necessitates a large off-momentum DA.
The paper is organized as follows: In Section II, the analytical expressions for the quadrupole strengths and other optics parameters of the TME cell are derived, includ-ing conditions for stability of the solutions and feasibility of the magnets. In Section III, the complete parametrization of the TME cells is performed using numerical examples of the analytical thin-lens solutions, applicable to the CLIC Pre-damping rings (PDR) lattice design. A validation of the method through the comparison of the results with numerical simulations using MADX [5] is presented in section IV. Finally, in section V an application of the analytical approach and the resonance free lattice concept [6] is used for the linear and non-linear optimization of the CLIC PDR.

II. ANALYTICAL APPROACH FOR THE TME CELLS
A. The TME cell A schematic layout of the TME cell is displayed in Fig. 1.  The horizontal emittance of the beam in an iso-magnetic ring: is determined by the average dispersion invariant in the dipoles, where α x , β x , γ x are the twiss parameters and D x , D x the dispersion and its derivative. The parameter C q = 3.84 × 10 −13 m is the quantum fluctuation coefficient for the electron, γ the relativistic factor, J x the damping partition number, and ρ x the bending radius. The minimization of the dispersion invariant average, provides the conditions of β x and D x at the center of the dipole, for achieving the theoretical minimum emittance [1]: where θ = l d ρx = 2π N d is the bending angle for N d dipoles in the ring. For a general TME cell, the geometrical emittance can be expressed as: where D xc and β xc the dispersion and beta functions at the center of the dipole. Substituting the values of D min xc and β min xc in Eq. (3) with their TME expressions of Eq. (2), the emittance becomes xTME =F C q γ 3 θ 3 . The scaling factor F for the TME lattice is F= 1 12 √ 15Jx and the damping partition number J x ≈ 1, in the case of isomagnetic rings, based on dipoles without quadrupole gradient [7]. Defining the ratios β r = βxc β min xc and D r = Dxc D min xc , it is useful to define the emittance detuning factor [2]: with x = r · x,TME . The detuning factor is an indication of how much the emittance deviates from its theoretical minimum, for a given set of optics parameters at the center of the cell.
Inverting Eq.(4) and solving with respect to β r , the following expression is computed: The quadratic dependence on D r of the argument in the square root in Eq. (5), sets an upper and a lower limit for the dispersion at the center of the dipole, in order for β r to be a real number: B. Analytical solutions for the quadrupole strengths The beta β xc and dispersion D xc functions, at the dipole center, impose two independent optics constraints and thus at least two quadrupole families are needed for achieving them. The horizontal optics functions are fully controlled by these two pairs of quadrupoles, whereas in the absence of additional knobs, the vertical plane optics is also uniquely defined.
Using basic linear optics arguments and the thin lens approximation and for specific β xc and D xc at the center of the dipole (or β r and D r ), analytical expressions can be derived for the strengths of the quadrupoles: which are parametrized with the drift lengths s 1 , s 2 , s 3 . The parameter D s is the dispersion at the center of the cell (between two mirror symmetric quadrupoles) and is a function of the drift lengths, the optics functions at the dipole center and the bending characteristics: where: The calculation of D s springs from the symmetry requirement at the middle of the cell, α x = 0. By applying the TME conditions at the middle of the dipole (α x =0, D x =0), the α x function at the middle of the cell has a quadratic dependence on (D −1 s ), which results in the two solutions, with opposite sign in the second component, for D s .
The horizontal and vertical phase advances of the cell can be defined through the trace of the cell transfer matrix and from this, the horizontal phase advance can be written in a simple form as: For D r = β r = 1, µ x = arccos(1/4) = 284.5 o independent on any cell parameter, which is a known property of the TME cells [2]. The expression for the vertical phase advance has a more complicated form: where L c = l d + 2(s 1 + s 2 + s 3 ) the cell length and s 23 = s 2 + s 3 . Unlike the horizontal plane, the vertical phase advance depends not only on the optics functions at the dipole center but also the cell geometry.

C. Momentum compaction factor
An analytical expression can also be derived for the momentum compaction factor of the cell, under the TME conditions (D x = 0 at the center of the dipole), and can be written in the form: depending only on the dipole characteristics and in particular, quite strongly on the bending angle, which explains the trend that the momentum compaction factor is reduced, when the dipoles become shorter and/or weaker. The momentum compaction factor for the absolute minimum emittance (D r = 1) is: which depends only on the dipole bending angle.

D. Optics stability
The stability criterion for both horizontal and vertical planes is: where M x,y is the transfer matrix of the cell and µ x,y are the horizontal and vertical phase advances per cell, respectively. The latest ensures the optics stability and can be used for constraining the cell characteristics (focal and drift lengths).
In the absolute minimum emittance limit, where β r = D r = 1, the parametric equations for the quadrupole strengths are reduced to: Applying the requirement of opposite sign quadrupole strengths in the above equations, thus f 1 × f 2 < 0, in order to assure optics stability in both planes, the case of (-) sign of Eq. (8) can be ignored. This shows that the dispersion at the symmetry point of the cell can never become negative. For the (+) sign, the following constraints are derived: It is interesting to study the behavior of Eqs. (15) in the limit where the drift spaces lengths are going to zero. They are then reduced to: In the limits where s 1 → 0 or s 3 → 0 both f 1 and f 2 converge to specific values, depending on the dipole length and on the drift spaces lengths. Thus, realistic solutions exist even if the first quadrupole Q 1 is placed exactly after the dipole, without any space between them, or if the two Q 2 quadrupoles are merged to 1. In the limit where s 2 → 0 both the focal lengths f 1 and f 2 go to zero or the quadrupole strengths to infinity. A good separation of the two quadrupoles is thus necessary in order to have a feasible TME cell. In the limit of the absolute minimum emittance and of s 2 → 0, the cos φ y function goes also to infinity verifying that those solutions are optically unstable.

F. Magnet technology constraints
Even if satisfied, the stability criteria do not necessarily guarantee technologically feasible magnet strengths. The pole tip field of the quadrupoles and chromatic sextupoles is constrained by the maximum value allowed by the chosen magnet technology. In addition, the radius of the magnets' aperture should be greater than a minimum value, defined by beam and lattice properties.
The quadrupole gradient (expressed in [T/m]) is defined as g = k(Bρ x ), where k the quadrupole strength and Bρ x the magnetic rigidity. From the definition of the pole tip field: where R is the quadrupole aperture radius. Considering a circular beam pipe, the minimum required aperture radius in order to accept all the particles of the incoming beam, for a non-Gaussian beam distribution, is defined by the displacement of the particles with the maximum action in the beam, defined by an emittance max and a momentum deviation (δp/p 0 ) max [8]: where β and D the beta and dispersion functions at this location, (δp/p 0 ) the total energy spread of the beam and d co a constant reflecting the tube thickness, mechanical tolerances and maximum orbit distortion. For a Gaussian beam distribution, Eq. (17) becomes: The R min can be computed for each element of the cell and takes its maximum value at the center of the quadrupoles, where the beta functions become maximum. The magnet technology constraint for the quadrupole gradient or strength is then: In a similar way, a magnet technology constraint can be set for the sextupole strengths. As already mentioned, the TME cells are intrinsically high chromaticity cells when targeting to their theoretical minimum emittance limit, as low dispersion and strong focusing are needed to achieve the ultra low emittance. The high chromaticity requires strong sextupoles for the chromaticity correction, reducing the dynamic aperture of the machine. The sextupoles used for the natural chromaticity correction are usually placed close to the quadrupoles, to large dispersion and beta function regions. In order to simplify the calculations, the sextupoles are considered to be placed on top of the quadrupoles, with equal lengths. The pole-tip field for the sextupoles is B s = (Bρ x )b 2 R 2 = 1 2 R 2 ∂ 2 By ∂x 2 | y=0 and the sextupole gradient (Bρ x )b 2 = B s /R 2 . As the sextupoles are set to cancel the chromaticity induced by the quadrupoles, the sextupole strengths can be calculated by: where K x,y the focusing and defocusing quadrupole strengths and S = b 2 (Bρx) the sextupole strengths. Evaluating the above integrals along the cell, the expressions for the sextupole strengths are: where ξ q x,y = − 1 4π β x,y K x,y ds and l q the length of the quadrupoles. For simplicity, we consider all the quadrupoles to have the same length. In the expressions above, the index f denotes the values of the optics functions on the focusing quadrupoles while d the values on the defocusing quadrupoles. In order to have feasible solutions, these values need to satisfy the constrain: Equations (5), (6), (7), (10), (18), (21) fully describe the linear optics of the TME cell.
The parameter space of the cell, including geometrical and optical properties, can be determined giving the possibility to optimize the cell according to any design requirements.

III. NUMERICAL APPLICATION
The analytical parameterization can be used to study the performance of any TME cell Having the drift lengths fixed, Eq. (7) combined with Eqs. (5) and (6) are studied numerically for different detuning factors r . In this example, the dipole bending angle is set to θ = 2π/38 and the drift lengths to s 1 =0.9 m, s 2 =0.6 m and s 3 =0.5 m. This configuration was found to be the optimal one for the CLIC PDR lattice design, as will be shown later.   In order to achieve the absolute minimum emittance, only one pair of initial optics functions (D xc , β xc ) or (D r , β r ) exists [2]. However, relaxing this requirement and detuning the cell to higher emittance values ( r > 1), several pairs of (D xc , β xc ) lying in elliptical curves can achieve the same emittance, as shown by equation (3). Fig. 5 (left) shows the solutions of (D r , β r ) color-coded with the detuning factor r . Even though, by definition, all solutions are stable in the horizontal plane, only a small fraction of them satisfy the stability criteria of the vertical plane (black squares). The parametrization of the focusing strengths with the emittance is displayed in Fig. 5 (right), with the same color-convention as before. The Scanning in a broader range of the detuning factor, two different types of solutions survive the stability criteria. Solutions with focusing Q 1 and defocusing Q 2 are presented in the top part of Fig. 6, while the opposite case is presented in the bottom. Following the convention of [10], we will refer to the former case as conventional TME cell while to the later as modified TME cell. The parametrization of the cell detuning factor r (left), and the horizontal (middle) and vertical (right) chromaticities with the horizontal and vertical phase advances of the cell is presented for each case. For a conventional TME cell the chromaticities get minimized in both planes towards small phase advances, while the emittance detuning factor gets large values. Large phase advances correspond to high chromaticity values and small detuning factors. It is interesting to notice that the high detuning factor solutions at large horizontal phase advances produce large chromaticities, as they correspond to minimum dispersion and beta functions at the center of the dipole which require strong focusing. In the case of the modified TME cell, the chromaticities are minimized for small phase advances as well, however in this case solutions with small detuning factors also exist. This type of cells is discussed in detail in [10]. The analytical solutions are shown in black, the solutions satisfying the stability criteria in red while the MADX solutions are presented in green. The agreement for the thin lens is excellent, demonstrating the validity of the analytical calculations. It is very interesting that, even in the thick lens case, the agreement is still very good. The analytical solution can be a very good approximation of the simulation results and can be helpful for the lattice optimization and understanding. In this way the optimal dipole characteristics, the geometrical characteristics of the cell and the interesting phase advances can be defined. It can also be very useful, for the definition of initial conditions to be used for the lattice design using numerical tools, whose optimization algorithms depend heavily on the initial values.

IV. VALIDATION OF THE METHOD
This approach was thus used in order to define the optimal configuration and working point of the TME arc cells of the CLIC PDR lattice design.

V. APPLICATION TO THE CLIC PDR DESIGN
The CLIC Pre-damping rings provide the first stage of damping of the e + /e − beams of the linear collider. They have to accommodate a 2.86 GeV beam with a large input emittance of 7 mm-rad, for positrons [11], and damp it down to a normalized emittance of 63 µm-rad for injection into the main DR. The required input and output parameters are given in Table I, for both electrons and positrons [12]. Unlike the DR, the PDR lattice design is not driven by the emittance requirements [9].

A. Non linear optimization
The main limitation of the DA in the low emittance lattices comes from the non-linear effects induced by the strong sextupole strengths, which are introduced for the chromaticity correction. From the non-linear dynamics theory [14], a resonance of order n defined by n x Q x + n y Q y = p, with |n x | + |n y | = n the order of the resonance and p any integer, is associated with a driving term. Based on [6], the driving term of a resonance associated with the ensemble of N c cells vanishes, if the resonance amplification factor is zero: e ip(nxµx,c+nyµy,c) = 1 − cos[N c (n x µ x,c + n y µ y,c )] 1 − cos(n x µ x,c + n y µ y,c ) = 0.
This is achieved if: N c (n x µ x,c + n y µ y,c ) = 2kπ, provided the denominator of Eq. (22) is non zero, i.e.: n x µ x,c + n y µ y,c = 2k π, with k and k any integers. From this, a part of a circular accelerator will not contribute to the excitation of any non-linear resonances, except of those defined by η x µ x + η y µ y = 2k 3 π, if the phase advances per cell satisfy the conditions: N c µ x = 2k 1 π and N c µ y = 2k 2 π, where k 1 , k 2 and k 3 are any integers. Prime numbers for N c , which in our case is the number of TME cells per arc, are interesting, as there are less resonances satisfying both diophantine conditions simultaneously.
The nonlinear optimization of the CLIC PDR lattice was based on the resonance free lattice concept, described above. From Eq. (4,0,0,0) and (3,1,0,0) are weakly excited with respect to the other modes. The vertical modes (0,0,4,0) and (0,0,3,1) are also excited, in the high horizontal phase advance limit for the first and in the high vertical phase advance limit for the second. All resonance driving terms are suppressed, for phase advances that are integer multiples of 1/17, as expected.
Here, the resonance driving terms are presented and discussed only to demonstrate the proof of principle of the resonance free lattice concept. In a further non-linear optimization of the lattice, especially when high-order magnet errors are included, additional families of sextupoles, in non-dispersive areas, can be used for the minimization of the resonance driving terms which limit the dynamic aperture.
Another quantity that has to be taken into account, is the amplitude dependent tune shift δq x,y /δJ x,y . From first order perturbation theory, the leading order tune shift can be represented by [15]: where, α ij are called the normalized anharmonicities and they describe the variation of the tune at different amplitudes (or action).  For the chromaticity correction, four families of sextupoles are used. A set of sextupoles are located before the focusing quadrupoles of the TME cells and a set of sextupoles after the defocusing ones. The same set-up is followed for the two other families of sextupoles, which are placed in the half TME cells of the dispersion suppressors. As those sextupoles are not placed in dispersive areas, they do not contribute to the chromaticity correction, but they can be used for further non-linear optimization of the lattice.

B. Dynamic aperture
The Dynamic aperture (DA) is defined as the maximum phase-space amplitude within which particles do not get lost as a consequence of single-particle effects [16]. The DA has to be at least equal or larger than the minimum beam transverse acceptance, R min . The beam coming from the positron source is not expected to be Gaussian, and the distribution in the storage ring is not modified, until the beam is damped close to equilibrium. For this reason, the minimum transverse acceptance is defined in terms of a maximum emittance max of the particles with the maximum betatron action in the beam, and of a maximum relative momentum deviation (δp/p 0 ) max [8]: The incoming beam to the CLIC PDR is a round beam with same horizontal and vertical rms emittances of rms x,y =7 mm-rad where, 99.9 % of the particles are inside a maximum emittance of max =10 rms x,y and with maximum (δp/p 0 ) max = 3%. Applying this to Eq. 24, the minimum acceptance can be calculated around the ring and is shown in Fig. 15 The DA of the ring was computed with numerical particle tracking, over 1000 turns, with the PTC module of MADX [5]. Fig. 16 shows the initial positions of particles that survived over 1000 turns, normalized to the horizontal and vertical beam sizes, at the point of calculation (σ x =4 mm, σ y =2 mm). The results for δp/p 0 = 0% are shown in red, for δp/p 0 = 1.2% in green and for δp/p 0 = −1.2% in blue. The minimum acceptance is shown in black. For these calculations the magnet fringe fields are taken into account, while any magnet error ing an optimization procedure based on the resonance free lattice concept, however, more optimization steps is required when magnet errors and the effect of wigglers are included. The frequency map analysis (FMA) examines the dynamics in frequency space rather than configuration space. Regular or quasi-regular periodic motion is a single point in frequency space characterized by a pair of fixed tune values. Irregular trajectories exhibit diffusion in frequency space, with the tunes changing in time. The mapping of configuration space (x & y) to frequency space (Q x & Q y ) will be regular for regular motion and irregular for chaotic motion. Numerical integration of the equations of motion, for a set of initial conditions (x, y, x , y ) and computation of the frequencies as a function of time (or turn number), constructs the map from the space of initial conditions to frequency or tune space, over a finite time span T [17][18][19]. An indication of how much the frequency is changing with time, is measured through the diffusion coefficient, defined by: where the index 1 refers to a certain number of turns, while, the index 2 to a consecutive same amount of turns. Large negative values of D denote long term stability while values of D close to zero denote chaotic motion [17].
Tracking of particles with different initial conditions for 1024 turns, was performed with MADX-PTC [20]. The ideal lattice including sextupoles and fringe fields is used, while no magnet errors are taken into account. The frequency map analysis was performed with the Numerical Analysis of Fundamental Frequencies (NAFF) algorithm [17]. Resonance lines in the frequency maps are shown as distorted areas, while the colors allow to relate the resonant features observed, to regions of the physical space [17]. From the frequency maps it is observed that the tune is crossing the (1,4) resonance, which is not eliminated by the TME phase advance choice (µ x = 5/17, µ y = 3/17) as shown in Fig. 12.
This seems to be the main limitation of the DA.
The shape of the frequency maps, especially at high amplitudes, does not have the tri-angular shape expected by the linear dependence of the tune shift to the action, and they appear to be folded. This occurs when terms of higher order in the Hamiltonian become dominant over the quadratic terms as the amplitude increases [17]. This behavior occurs due to the suppression of the lower order resonances, following the resonance free lattice concept, which gives rise to higher order terms. Even though folded maps may lead to potentially very unstable designs, in our case this is not taken into account for the moment, as the folding of the map appears at high amplitudes, beyond the DA aperture limit.

VI. CONCLUSION
An analytical parametrization for the TME cell has been derived and presented in this paper, based on linear optics arguments and the thin lens approximation. In that way all cell properties, optical and geometrical, are globally determined and the optimization procedure following any design requirements can be performed in a systematic way. Stability criteria in both horizontal and vertical planes and magnet technology constraints are also applied.
A comparison of the analytical solution with the results from the simulation code MADX gave very good agreement, even for the thick-elements optics. This method provides a very useful tool for defining optimal regions of operation for the best performance of the cell, according to the requirements of the design. The analytical approach and the resonance free lattice concept were finally used for the linear and non-linear optimization of the CLIC Pre-damping rings, providing an adequate dynamic aperture for a large incoming beam.