The Mathieu unit cell as a template for low emittance lattices

The multi-bend achromat (MBA), which often serves as a building block for modern low-emittance storage rings, is composed of a repetition of unit cells with optimized optical functions for low emittance in the achromat center, as well as end cells for dispersion and optics matching to insertion devices. In this work, we describe the simplest stable class of unit cells that are based on a longitudinal Fourier expansion, transforming Hill equations to Mathieu equations. The resulting cell class exhibits continuously changing dipolar and quadrupolar moments along the beam path. Although this elementary model is defined by only three parameters, it captures a significant amount of notions that are applied in the design of MBAs. This is especially interesting as Mathieu cells can be viewed as an elementary extension of Christofilos' original model of alternating-gradient focusing. Mathieu cells can be used to estimate the range of reasonable cell tunes and put an emphasis on the combination of longitudinal gradient bending and reverse bending, as well as on strong horizontal focusing to reach emittances lower than the classic theoretical minimum emittance cell. Furthermore, the lowest emittances in this model are accompanied by small absolute momentum compaction factors.


I. INTRODUCTION
For practical reasons, the treatment of accelerator lattices is mostly based on modeling them with discrete elements representing accelerator magnets. However, the evolution of multi-bend achromats (MBAs) has shown that longitudinal gradients in magnet strength can significantly decrease emittance (e.g. [1][2][3]), that reverse bends [4] (see also Veksler's suggestion in [5]) are necessary to fully exploit these longitudinal gradients [6,7] and that combined-function magnets can help to decrease emittance by manipulating damping partitions [8].
These facts can inspire to model the focusing and bending functions of the periodic lattice structure (i.e., the unit cell) directly, by a set of basis functions that are periodic in cell length, instead of using distinct elements to represent magnets. In principle, the type of basis function can be selected in an arbitrary manner. E.g., in [9] step-like basis functions are used and truncated at a high order, and a particle-swarm based optimization is applied in the resulting high-dimensional parameter space.
The choice of sinusoidal basis functions is motivated in Sec. II -when keeping the amplitude of a given multipole on the beam path constant, higher harmonics of the unit cell require stronger magnet pole-tip fields than lower harmonics.
The focusing functions for Mathieu cells, which we introduce in this work, are discussed in Sec. III, contain the lowest possible order of such basis functions that yield stable solutions. It is interesting to note that these sinusoidal focusing forces are also the starting point for Christofilos' description of alternating-gradient focusing [10]. However, his derivations focus on qualitative aspects of the motion, and not on solving the underlying differential equations -these are Mathieu equations.
Afterwards, bending functions are included in Sec. IV. For the resulting cells, synchrotron integrals, emittance and momentum compaction can be computed, and example solutions are studied.
The scaling laws for unit cells are investigated in Sec. V with an emphasis on the 'chromaticity wall' and selecting the optimal cell length. A new objective function for the emittance of an arc with optimally scaled cell lengths is obtained, including constraints on applicable sextupole field strength. After further approximating the applicable poletip fields of magnets for a specific example tune, an example cell is constructed using parameters of the SLS 2.0 storage ring in Sec. VI.

II. LONGITUDINAL HARMONICS
Consider the magnetic field on a cylinder with variable radius r , and the beam path leading through its axis. For simplicity, we neglect the curvature of the path, although the argument naturally extends to that case. In a currentfree region, a scalar potential defining the magnetic field B = −∇V obeys the Laplace equation ∇ 2 V = 0 [11]. In the aforementioned periodic cell, its solution can be expressed as wherek p = 2πp/L, and I m is the modified Bessel function of the first kind and order m. V n,p is a Fourier representation of the 2n−polar fields on the beam path. When selecting only a single Fourier component with fixed n = 0 and arbitrary p, the radial field component at radius r is given as For p = 0 this reduces to the commonly known behavior B r ∝ r n−1 . However, the higher the longitudinal harmonic arXiv:2005.04087v1 [physics.acc-ph] 8 May 2020 |p| to be considered, the more difficult an application of the desired on-axis multipolar fields will become. Therefore, lower longitudinal harmonics of multipolar fields are preferable to higher harmonics. Further assuming the unit cell to possess symmetry planes, we can select cosine functions cos(k p s) as basis functions with increasing positive order p ≤ P .

A. Biplanar stability
The next task is finding the lowest maximum order P for which stable particle motion could be achieved. The transverse linear motion of a charged particle in a decoupled accelerator lattice without bending magnets can be described using Hill differential equations [12,13] Assuming κ(s) to be constructed of basis functions cos(k p s), the most elementary case to consider is P = 0 because then κ = const. As the sign of κ is different for the horizontal and vertical plane, bounded motion can only be achieved in one of them, and stable particle motion is impossible.
On the other hand, as we will see, the case P = 1 already allows for stable motion. The resulting parameter space is low-dimensional, so it can be fully explored. We first investigate such a model without bending and thus without dispersion. An additional parameter for bending is then included, and synchrotron radiation integrals (including damping partition, emittance, momentum compaction) are computable.

III. MATHIEU EQUATIONS IN 2D
To simplify the following calculations, we consider a normalized cell with the dimensionless length π. The normalized longitudinal cell coordinate u is linked to the standard cell coordinate via s = Lu/π (see Appendix).
Still considering the aforementioned focusing function for the case P = 1, we obtain where the factor −2 was selected arbitrarily for alignment with standard notation. The equations of motion are now Mathieu equations, both depending on the same set of parameters k 0 , k 1 . We analyze the horizontal motion based on Floquet solutions, mainly following the approach outlined in [14]. These can be written in the normal form [15] X (u) = e i2ν x u f (u), (6) where 2ν x is the characteristic exponent and ν x is the horizontal cell tune, i.e., the betatron phase advance in a cell divided by 2π. We express the π-periodic function as a truncated Fourier series with the highest harmonic being Q. For the following calculations, Q = 50 is sufficient.
It is apparent that periodic solutions only exist for leading to limited regions in (k 0 , k 1 ) space where horizontally stable motion occurs. Furthermore, it follows from Eq. (4) that stability of vertical motion is equivalent to that of horizontal motion when mirroring the (k 0 , k 1 ) regions at the origin. The intersection of stability regions for both planes leads to islands of stability for transverse motion (see Fig. 1, cf. [16,Fig. 5]). The islands differ significantly in the maximal focusing strength that needs to be applied. The only stable solutions with reasonable max |k(u)| ≤ 2 all occur in a single stability island. This 'neck-tie' island, named here in analogy to the corresponding diagram for the FODO lattice [17] is shown in Fig. 2 in more detail. We conclude that reasonable cell designs require (k 0 , k 1 ) in this island, which has cell tunes ν x , ν y < 1/2.

A. Tune map for chromaticity
As there exists a bijective mapping of stable-motion quadrupole configurations to tunes (k 0 , k 1 ) ↔ (ν x , ν y ), we are able to study the properties of Mathieu unit cells directly in tune space.
Given ν x , one may solve Eq. (8) for f (u). Optical functions are computed from X (u) in Eq. (6) as (see e.g. [18]) with and they can be used to compute the linear chromaticity with the horizontal and vertical optical functionsβ x ,β y by (cf. [19]) Following from the aforementioned symmetry of vertical and horizontal motion in (k 0 , k 1 ) space, we obtain the vertical chromaticity for a given tune as The results of the linear chromaticity computation are shown in Fig. 3. In the usable regions of the tune map, i.e., considering stop-bands around the half-integer resonances, we obtain negative chromaticities ξ x,y > −2.5.
It should be noted that without chromaticity compensation, the general dependency of cell tune on particle energy can be obtained by scaling the (k 0 , k 1 ) vector corresponding to a given tune in the neck-tie diagram in Fig

IV. BENDING AND EMITTANCE
The next task is to include bending into the unit cell. We assume that the curvature is sufficiently small so that we can neglect the effect of weak focusing on k(s) in Eq. (4). Following the same line of reasoning we also neglect edge focusing.
The average curvature 1/ρ in a cell is defined by the arc geometry of the storage ring and the bending of endcells. When assuming the curvature to contain low-order longitudinal harmonics in the same manner as the focusing strengths (P = 1), we can parameterize It should be noted that an upper limit on |b|, and thus |b 1 |, exists given by achievable dipole field strength independent of cell length, as Here we introduced the characteristic magnetic field density depending on the beam rigidity (B ρ).
Normalizing with the average curvature, the inhomogeneous Hill equation for linear dispersion η(s) [17,19] can be rewritten as (see Appendix) We recognize that the solutionsη(u) of Eq. (21) are additive in b(u). Letη (0) (u) be the solution for b(u) = 1 and let η (1) (u) be the solution for b(u) = −2 cos(2u). Then the general solution is linear in b 1 , as The driving term b(u) requiresη(u) to be periodic in π, so that Eq. (21) reduces to the solvable linear equation system ) and all other components of c zero. The solutionη(u) can then be constructed using v.

A. Synchrotron integrals
Having introduced bending and dispersion, knowledge of linear momentum compaction can be obtained, which is proportional to the synchrotron integral [20,21] for the normalized cell. To gain some insight into the behavior of I 1 , we insert Eq. (23) and obtain By its definition preceding Eq. (23) and due to symmetry conditions, π 0η (1) du = 0.
Althoughη(u) is the solution of a driven parametric oscillator, we may expect it to mainly oscillate at the driving frequency cos(2u), making the last coefficient in Eq. (27) small. We proceed by computing radiation properties for the normalized cell. The synchrotron integrals related to radiation loss and damping partitions are [20,21] and The expression used for I 4 is an approximation in which, in consistence with our assumption, the contribution of weak focusing has been omitted. In full analogy to I 1 and substitutingη (·) → kη (·) , we find that I 4 (b 1 ) is also a quadratic function of b 1 .
In order for a flat lattice to allow damping in all dimensions, the horizontal damping partition [20,21]. In low-emittance rings, J x > 1 is favored [22] as the effects of quantum excitation are then shifted from the transverse into the longitudinal plane.
The dispersion action H (s) occuring in the quantum excitation integral can be computed using the Floquet solution as One can then obtain the emittance ∝ I 5 /(I 2 J x ). However, we are interested in the emittance relative to that of a normalized theoretical minimum emittance (TME) cell [23], as it is independent of cell length.

B. Results
We can now search for the optimal b 1 parameter to reach minimum emittance ratio F for a given tune (ν x , ν y ); the results are shown in Fig. 4. Sub-TME emittances are reached for 0.4 < ν x < 0.5, with a minimal F < 0.7. We see that, in this band, increasing ν y only has slight effects -increasing J x and decreasing F . Damping partitions for the sub-TME region are in a feasible interval J x ∈ [1.5, 2.5].
The region with small absolute momentum compaction in Fig. 4 has a similar location and shape as that of sub-TME emittance -this is consistent with the general observation that low-emittance lattices require small absolute momentum compaction.
To further investigate the influence of the parameter b 1 , which is not visible in the projections in Fig. 4, figures of merit for an example tune ν x = 0.45, ν y = 0.35 and variable dipole coefficient are shown in Fig. 5. According to Eq. (27) we expect I 1 to be quadratic in b 1 , with the quadratic coefficient almost vanishing -we obtain a visibly linear dependency here. The location of I 1 = 0 and the location of the minimal F again illustrate that low emittances and low momentum compaction are closely related.
As the damping partition is in a usable range, the minimum emittance solution for this tune is feasible. The example solution parameters, figures of merit, and optical functions are shown in Fig. 6 and Table I. It can be seen that (1) positive bending and defocusing quadrupole fields overlap, increasing J x [8], and that (2) reverse bending occurs at the position of maximum dispersion [6].
For the unit cell with length π, this results in an equation system  Table I. Bottom: corresponding distribution of dipole (black), quadrupole (blue) and sextupole fields (yellow). sextupole coefficients for F -optimized cells with given cell tunes are shown in Fig. 7 and also included for the example in Table I.

A. Sextupole-limited arc emittance
We want to find the cell length yielding the optimal emittance for a given limited sextupole strength max |m|. In an arc of constant average curvature 1/ρ , the actual sextupole strength µ(u) scales relative to the sextupole strength of the normalized cell m(u) as (see Appendix) This disadvantageous dependency on cell length is sometimes referred to as 'chromaticity wall' [24] and is a major limitation for shrinking unit cells.
The optimal cell length can be obtained from the above equation as It is well known (e.g. [25]) that the emittance scales with the cube of bending angle per cell, and thus in our case ∝ L 3 .
Reusing the definition of TME-normalized emittance F in Eq. (34), we find that the optimal emittance scales as We can use G as an objective function for optimization, thus including sextupolar fields in a straightforward manner, to find an optimal value for b 1 .
Tune maps for figures of merit in which b 1 is selected to yield the optimal G are shown in Fig. 10. We can observe that the characteristics for the emittance ratio F and the damping partition J x did not change significantly, although the tune-space region of low momentum compaction has reduced in size.
Furthermore, it is interesting that the two regions with G ≤ 1 exist. One region has a significantly reduced horizontal focusing ν x < 0.2. Unfortunately, the low G values in this region are mainly influenced by a large and infeasible damping partition J x > 3 (see Fig. 8).
The other region overlaps with the low-emittance regime shown in Fig. 4, with the difference that there is now a slight preference for less vertical focusing. The additional parameters G and max |m| for our example configuration, which is located in that region (see also Table I and Fig. 5), are shown in Fig. 9.

B. Extensions to higher harmonics
Having obtained an optimized solution for the Mathieu cell (P = 1), it is possible to iteratively increase P and reoptimize the solution locally. However, the number of free parameters increases significantly. In the scope of this work, we increase to P = 2 only for the sextupolar field component, so that m(u) = m 0 + 2m 1 cos(2u) + 2m 2 cos(4u).
This has the advantage that the dimensions of the free parameter space (ν x , ν y , b 1 ) do not increase -the additional harmonic coefficient m 2 is used to reduce max |m| without changing optical functions.
To compensate chromaticity, we are required to solve a more general variant of Eq. (36) with the components of A x,y holding scaled Fourier components ofηβ x,y . This system is underdetermined; its solution space in three dimensions is given as with m (0) being an arbitrary solution. For our computation we use the least-squares solution of the system (41). The quantity max |m| can be computed with minor effort, as we require it to be minimal under the constraint of full chromaticity compensation -this is achieved using an elementary optimization procedure on the scalar a.
The results of this optimization in tune space are shown in Fig. 12. Relative to the setup using just constant and fundamental harmonic (P = 1), an overall reduction of the G objective has been achieved, reaching values G < 0.7 in the low-emittance region.
This can be observed in more detail for our example tune ν x = 0.45, ν y = 0.35 in Figs. 9 and 11. The maximum value of |m(u)| has been reduced by decreasing the sextupole strength at the position of maximum bending. This is reasonable as the large sextupolar fields at this location have a negligible influence on chromaticity compensation.
According to Eq. (19) and assuming a normal-conducting magnet limit of max B ∼ 2 T, we get max |b| ∼ 6.2, or max |b 1 | ∼ 2.6. We assume the maximum applicable sextupole strength at max |µ| = 650 m −3 , which is a conservative estimate consistent with the present lattice design. By using Eq. (38) we are able to compute the optimal cell length for a Mathieu cell with example parameters for SLS 2.0. Using the standard sextupole harmonics (P = 1, Table I) we obtain max |m| ∼ 1.726, resulting in an optimal cell length of ∼ 1.592 m. Using the extended sextupole harmonics (P = 2, Fig. 9) we obtain a reduced value of max |m| ∼ 1.134, resulting in an optimal cell length of ∼ 1.433 m.

A. Improved optimal cell length estimate using pole-tip fields
It should be noted that, due to the overlapping of fields with different multipolar order, the pole tip field of a combined-function magnet will be higher than that of the sextupole component, thus increasing the optimal cell  Fig. 9. The optical functions are identical to those in Fig. 6. length. For a detailed example we calculate pole-tip fields B pt r with the common approach [26], i.e., without considering longitudinal variation as in Eq. (1), as with the pole-tip radius R, or as a unitless equation, where we defined the characteristic length containing the geometric mean of chamber and average curvature radius. In the case of interest, the maximum poletip field strength is not dominated by m(s) alone, as would be the case for L c /L 1. Instead, the situation L c ∼ L occurs because multipoles of different order often have comparable pole-tip field magnitudes. While the pole-tip field can be used as an estimate for the technical feasibility of magnet design, this estimate can be improved further. To do so, we take into account the empirical knowledge that the feasible pole-tip fields decrease with the multipole order n -e.g., for the SLS 2.0 separate-function magnets we may assume an inverse relation max B with their order, leading to the definition of a weighted poletip field via We can obtain good approximations of the maximum pole-tip fields for a given value of L c /L by computing the maximum value of B r /B c on a grid of (φ, u) points. In this work we use 128 values of u and 16 values of φ. The result of this computation with the example cell is shown in Fig. 13. One can observe that, as expected, the sextupole strengths dominate for large L c /L; small values are dominated by the constant dipole contribution.

B. SLS 2.0 parameters and results
For SLS 2.0 we assume a chamber radius R = 10 mm and obtain the characteristic length L c ∼ 1.565 m. The technical limit of pole-tip fields in such a distributed magnet structure is yet to be determined. Comparing the actual pole-tip field in Fig. 13 with the sextupole-only contribution, we can see that the optimal cell length increases significantly when all multipoles are considered.
We now consider the example values marked in Fig. 13, where the optimal cell length is L ∼ 1.031L c ∼ 1.614 m and max |B w r | is close to 2 T with a small safety margin. The distribution of multipole contributions to the pole-tip fields is shown in Fig. 14.
This example magnet configuration is analysed using the optics code OPA [26]. As optics codes usually do not work in Fourier space, we discretize the solution into segments of dipole-quadrupoles and thin sextupoles. For convenience, we choose 128 segments for each magnet type.
The optics results are shown in Fig. 15, and Table II shows global figures of merit as computed by OPA. For the betatron tunes, we can observe that for our example, neglecting weak focusing and edge focusing as stated in Sec. IV is justified. Within the assumptions about pole-tip fields, which may exceed technical limits, and our assumptions about weak focusing and general feasibility of the non-trivial magnetic field arrangement, we obtain an emittance of ∼ 33.2 pm, which is significantly less than the SLS 2.0 design [27]. In addition to the aforementioned complications, the cell is   Fig. 15).
almost isochronous with a momentum compaction in the 10 −6 range. This can be circumvented by a minor decrease of b 1 at the expense of slightly increased emittance (see Fig. 5).

VII. CONCLUSION
In this work, we introduced Mathieu unit cells as elementary approximations for periodic lattice systems. Due to their distributed multipolar structure, they allow for the inclusion of combined-function effects, as well as the computation of common figures of merit like momentum compaction and emittance. They even predict the usefulness of combining longitudinal gradients with reverse bending. Of course, this work can only be a minor step in the direction of more complete studies, in the line of what has been performed e.g. for damping rings [25,28]. Robust designs require a more detailed analysis of the nonlinear properties, see e.g. [29,30].
Although their distributed multipolar fields would require special-purpose magnets -probably something similar to Canted Cosine Theta technology [31] -Mathieu cells are useful tools for investigating basic lattice configurations and performance limits. Taking into account further progress on MBA miniaturization and combined-function magnet lattices, e.g., [31][32][33][34], an understanding of the properties of Mathieu cells could help future applications.
The source code for all computations in this work, excluding the ones performed in OPA, is based on the SciPy framework [35,36] and fully accessible [37]. Since we require the tune for all cells to be independent of the cell length, this should also apply to the natural chromaticity so that βκ ds ∝ β/L is constant, and is linear in L, so β k ds is also constant, withβ being the optics function of the normalized unit cell. Furthermore, the dispersion function η(s) must fulfill the inhomogeneous Hill's equation Note that this inverse quartic scaling is due to the average curvature 1/ρ remaining constant -if the ring was miniaturized as a whole, 1/ρ ∝ 1/L would hold, resulting in inverse cubic scaling and corresponding to the multipole order.