# Emittance reduction with variable bending magnet strengths:
Analytical optics considerations and application to the Compact Linear
Collider damping ring design

S. Papadopoulou  European Organization for Nuclear Research (CERN), BE
Department, 1211 Geneva 23, Switzerland  Department of Physics, University of
Crete, Greece  Institute of Theoretical and Computational Physics (ITCP),
Crete, Greece  F. Antoniou  European Organization for Nuclear Research (CERN),
BE Department, 1211 Geneva 23, Switzerland  Y. Papaphilippou  European
Organization for Nuclear Research (CERN), BE Department, 1211 Geneva 23,
Switzerland

###### Abstract

One of the main challenges of the lattice design of synchrotrons, used as
light sources or damping rings (DRs), is the minimization of the emittance.
The optimal lattice configurations for achieving the absolute minimum
emittance are the theoretical minimum emittance (TME) cells. This paper
elaborates the optimization strategy in order to further reduce the betatron
emittance of a TME cell by using dipoles whose magnetic field varies
longitudinally. Based on analytical results, the magnet design for the
fabrication of variable bends with the optimal characteristics is discussed.
In order to have a global understanding of all cell properties, an analytical
approach for the theoretical minimum emittance (TME) cells with variable bends
is elaborated. This approach is employed for the design optimization of the
Compact Linear Collider (CLIC) DRs. The margin gained in the emittance
including IBS based on this new design strategy enables the removal of a
number of TME cells from the existing arcs while still keeping the
requirements of the collider. The reduction of the circumference is further
enhanced by the use of optimized high-field wigglers. The optimization
strategy followed for the CLIC DRs is explained in detail and the output
parameters of the new design are presented.

###### pacs:

##  I Introduction

The principle objective of linear collider damping rings (DRs) and storage
ring X-ray sources is the generation of ultra-low emittances in order to
achieve high brightness beams. The optics design is primarily focused on
building a compact ring, attaining a sufficiently low emittance and an
adequately large dynamic aperture (DA). Due to their compactness and the very
small horizontal emittance they reach, the TME cells Teng (1985); Einfeld  _et
al._ (1996) are preferred for some ring designs. The performance of these
cells was previously studied analytically for the case of uniform dipoles
Antoniou and Papaphilippou (2014). The emittance reached by a TME cell can be
further reduced if instead of standard dipoles, longitudinally variable bends
Guo and Raubenheimer (2002); Nagaoka and Wrulich (2007) are used. This paper
summarizes analytical and numerical studies regarding the variable bends that
provide emittances lower than the ones of a uniform dipole for a TME cell. The
optimal magnetic field evolution of these bends is found analytically and
applied to two simple dipole profiles (Section III). The characteristics of
the dipole with the optimal field variation are further refined by the
technological limitations for the magnet fabrication (Section IV). These
limitations are used as constraints in order to study the impact of each
dipole profile on the optics functions of the cell and on the properties of
the ring. The analytical parametrization of the quadrupole strengths and
optics functions with respect to the drift lengths and the emittance is
extended to the TME with a non-uniform dipole (Section V). The strong focusing
needed for accomplishing the TME conditions results in cells with high
chromaticities. In favor of having low chromaticity, a sufficient large
dynamic aperture and a compact ring, the TME should be detuned from the
absolute minimum emittance solution.

Based on the analytical thin-lens solutions, appropriate initial conditions
are chosen for matching the lattice numerically using MADX MAD-X homepage .
This approach is applied to the CLIC DRs, filled with longitudinally variable
bends in the arcs and high field wigglers in the long straight sections. The
optimization strategy followed to reduce the circumference of the DRs design
is explained in Section VI. To this end, the parameters of the new alternative
design are compared with respect to the original ones.

##  II Analytical solutions for minimizing the emittance of a TME cell

The equilibrium horizontal emittance in a storage ring is given by:

$\epsilon_{x}=\dfrac{C_{q}\gamma^{2}}{J_{x}}\
\dfrac{\langle\dfrac{\mathcal{H}}{\left|\rho\right|^{3}}\rangle}{\langle\dfrac{1}{\rho^{2}}\rangle}=\dfrac{C_{q}\gamma^{2}}{J_{x}}\
\dfrac{\dfrac{1}{C}{\int\limits}_{0}^{C}\dfrac{\mathcal{H}}{\left|\rho\right|^{3}}ds}{\dfrac{1}{C}{\int\limits}_{0}^{C}\dfrac{1}{\rho^{2}}ds}\;\;,$
(1)

where $C$ is the circumference of the ring, $\gamma$ is the Lorentz factor,
$J_{x}$ is the damping partition number and $C_{q}=3.84\times 10^{-13}$ m (for
electrons). The lattice function ${\cal H}$ known as the dispersion invariant,
depends on the optics parameters, the dispersion function and its derivative:

${\cal
H}(s)=\gamma(s)\eta(s)^{2}+2\alpha(s)\eta(s)\eta(s)^{\prime}+\beta(s){\eta(s)^{\prime}}^{2}\;\;.$
(2)

In the case of uniform dipoles, having a constant bending radius $\rho$, the
minimum emittance value is obtained through the minimization of the
$\left\langle{\cal H}\right\rangle$. However, in the case of longitudinally
variable bends, for a varying $\rho$ along the length of the magnet, the aim
is to minimize $\left\langle{\cal H}/\rho^{3}\right\rangle/\left\langle
1/\rho^{2}\right\rangle$.

A schematic layout of the TME cell is displayed in Fig. 1. It consists of one
dipole D of length $L$ and of two quadrupole families Q1, Q2 with focal
lengths $f_{1}[m]=1/(k_{1}l_{q_{1}})$ and $f_{2}[m]=1/(k_{2}l_{q_{2}})$, where
$k_{1},k_{2}$ are the normalized quadrupole strengths and
$l_{q_{1}},l_{q_{2}}$ their lengths. The drifts between the elements are
denoted by $s_{1}$, $s_{2}$ and $s_{3}$.

Figure 1: Schematic layout of a TME cell

For simplicity, the center of consecutive dipoles is considered as the
entrance and exit of the TME cell, with the index “cd” (center of dipole)
representing functions at this location. The optics parameters, the dispersion
function and its derivative from the center to the edge can be written as:

$\beta\left(s\right)=\beta_{cd}-2\alpha_{cd}s+\gamma_{cd}s^{2},~{}\alpha\left(s\right)=\alpha_{cd}-\gamma_{cd}s,~{}\gamma\left(s\right)=\gamma_{cd},~{}\eta\left(s\right)=\eta_{cd}+\eta^{\prime}_{cd}s+\tilde{\theta}\left(s\right),~{}\eta^{\prime}\left(s\right)=\eta^{\prime}_{cd}+\theta\left(s\right)\;\;.$
(3)

The expressions in Eq. (3) are used to calculate the dispersion invariant
${\cal H}(s)$ (Eq. (2)). It should be mentioned that the focusing of the
dipole is negligible for dipoles having a small bending angle $\theta$.

The theoretical minimum emittance can be achieved if the symmetry condition,
for which both beta ($\beta_{x}$) and dispersion ($\eta_{x}$) functions have a
minimum at the center of the bending magnets
($\alpha_{cd}=\eta^{\prime}_{cd}=0$), is satisfied (Fig. 2) Teng (1985);
Einfeld  _et al._ (1996).

Figure 2: Symmetry condition for the TME.

For isomagnetic TME cells the dispersion and beta functions at the center of
the dipole are respectively equal to $\eta_{cd}=\frac{\theta L}{24}$ and
$\beta_{cd}=\frac{L}{2\sqrt{15}}$. These functions are different for a non-
uniform dipole, since their bending angle and radius vary along the electron
beam path in the magnet. In that case, the bending angle and its integral are
given by:

$\theta={\int\limits}_{0}^{s}\dfrac{1}{\rho(s)}ds~{},~{}~{}\tilde{\theta}={\int\limits}_{0}^{s}\left({\int\limits}_{0}^{s}\dfrac{1}{\rho(s)}ds\right)ds\;\;.$
(4)

The beta and dispersion functions at the dipole center ($\beta_{cd}$ and
$\eta_{cd}$) impose two independent optics constraints, therefore, at least
two quadrupole families are needed for achieving them. Using the thin-lens
approximation and for given $\beta_{cd}$ and $\eta_{cd}$ at the center of the
dipole, the analytical expressions for the quadrupole focal lengths can be
derived:

$f_{1}=\dfrac{s_{2}g_{1}}{g_{1}-\eta_{ss}+s_{2}\theta}\;\;,\quad
f_{2}=\dfrac{s_{2}\eta_{ss}}{g_{1}-\eta_{ss}}\;\;,$ (5)

where the dispersion $\eta_{ss}$ at the center of the cell, between two mirror
symmetric quadrupoles,

$\eta_{ss}=\dfrac{\dfrac{-2g_{1}s_{3}}{s_{2}}}{1\pm\sqrt{1+\dfrac{4g_{1}s_{3}}{s_{2}^{2}}\dfrac{\beta_{cd}^{2}\theta-\left(L/2+s_{1}\right)g_{2}}{\beta_{cd}^{2}\theta^{2}+g_{2}^{2}}}}\;\;,$
(6)

depends on the drift lengths, the optics functions at the dipole center, the
bending characteristics and the parameters

$g_{1}=\eta_{cd}+s_{1}\theta+\tilde{\theta},\quad
g_{2}=\eta_{cd}-\dfrac{L}{2}\theta+\tilde{\theta}\;\;.$ (7)

In the limit of $s_{2}\to 0$, i.e. when Q1 and Q2 are merged into one
quadrupole, both $f_{1}$ and $f_{2}$ vanish, giving infinite focusing
strengths (see Appendix A). When $s_{3}\to 0$, the two quadrupoles that belong
to the Q2 family of the TME cell are merged into one and $f_{2}$ vanishes. In
this respect, ultra-short drift space lengths between the quadrupoles must be
avoided. The fact that in the limit of $s_{1}\to 0$ both $f_{1}$ and $f_{2}$
have fixed non-zero values, implies that it is possible to place the
quadrupole Q1 exactly next to the dipole, with no drift space between them.

Longitudinally variable dipoles, whose magnetic field varies along their
length, can provide lower horizontal emittances than a uniform dipole of the
same bending angle Guo and Raubenheimer (2002); Nagaoka and Wrulich (2007). In
the case of a TME cell, the evolution of ${\cal H}(s)$ along a uniform dipole
is shown in Fig. 3. This evolution guides the bending radius choice for
achieving an emittance reduction by using a variable bend. In fact, the
variable bend should be designed such that the minimum bending radius is at
the dipole’s center and then it should decrease towards the edge of the dipole
Papaphilippou and Elleaume (2005); Papaphilippou  _et al._ (2005); Wang  _et
al._ (2011); Wang (2009); Papadopoulou ; Streun and Wrulich (2015);
Papadopoulou  _et al._ (2015); Liuzzo  _et al._ (2016).

Figure 3: The evolution of the dispersion invariant along the uniform dipole.

For the present work, two different functions of the bending radius are used,
being either constant or linearly dependent with respect to the distance $s$.
Due to the optics symmetry, all definitions can be given for the half-dipole
(from 0 to $L/2$). In this respect, the first part of the half dipole at the
dipole center ($s=0$) has a length $L_{1}$ and the second, with a length
$L_{2}$, follows until the exit of the dipole, with functions $\rho_{1}(s)$
and $\rho_{2}(s)$, describing the bending radii in the respective parts.

The maximum magnetic field is at the center of the dipole, where the bending
radius is minimum. The minimum magnetic field and maximum bending radius are
at the edges of the magnet. The field evolution along the magnet can be well
described by using only two parameters; the lengths and the bending radii
ratio Wang (2009); Papadopoulou ; Papadopoulou  _et al._ (2015), that are
defined as:

$\lambda=\dfrac{L_{1}}{L_{2}}~{}~{}~{}\mathrm{and}~{}~{}~{}\tilde{\rho}=\dfrac{\rho_{1}}{\rho_{2}}\;\;.$
(8)

The lower $\lambda$ is, the shorter is the high field region compared to the
low field one. The lower $\tilde{\rho}$ is, the higher is the field at the
dipole center compared to the one at the edges. Generally,
$\tilde{\rho}<1~{}\mathrm{because}~{}\rho_{2}~{}\mathrm{needs~{}to~{}be~{}larger~{}than}~{}\rho_{1}~{}\mathrm{and}~{}\lambda>0$.

Splitting the half dipole in two parts with different bending radii requires
the study of the dispersion invariants for each part separately. In this
respect, the dispersion invariant given in Eq. (2) should be calculated for
both parts; that is ${\cal H}_{1}(s)$ and ${\cal H}_{2}(s)$ with:

${\cal
H}_{1,2}(s)=\gamma_{1,2}{\eta_{1,2}}^{2}+2\alpha_{1,2}\eta_{1,2}\eta^{\prime}_{1,2}+\beta_{1,2}{\eta^{\prime}_{1,2}}^{2}\;\;.$
(9)

After implementing the symmetry condition in Eq. (3), the optics functions for
the first and the second part of the half dipole are given by Eq. (10) and Eq.
(11) respectively, for
$\beta_{L_{1}},~{}\alpha_{L_{1}},~{}\gamma_{L_{1}},~{}\eta_{L_{1}}$ and
$\eta^{\prime}_{L_{1}}$ corresponding to the optics functions at the point
where $s=L_{1}$.

$\beta_{1}=\beta_{cd}+\gamma_{cd}s^{2},~{}\alpha_{1}=-\gamma_{cd}s,~{}\gamma_{1}=\gamma_{cd},~{}\eta_{1}=\eta_{cd}+\tilde{\theta}_{1},~{}\eta^{\prime}_{1}=\theta_{1}\hskip
11.38092pt(\alpha_{cd}=0,\eta^{\prime}_{cd}=0)$ (10)
$\displaystyle\beta_{2}=\beta_{L_{1}}-2(s-L_{1})$
$\displaystyle\alpha_{L_{1}}+(s-L_{1})^{2}\gamma_{L_{1}},~{}\alpha_{2}=\alpha_{L_{1}}-(s-L_{1})\gamma_{L_{1}},~{}\gamma_{2}=\gamma_{L_{1}},~{}$
(11) $\displaystyle\eta_{2}$
$\displaystyle=\eta_{L_{1}}+\tilde{\theta}_{2}+(s-L_{1})\eta^{\prime}_{L_{1}},~{}\eta^{\prime}_{2}=\theta_{2}+\eta^{\prime}_{L_{1}}\;\;,$

where the bending angles and their integrals, using Eq. (4), are expressed as:

$\theta_{1}={\int\limits}_{0}^{s}\dfrac{1}{\rho_{1}(s)}ds,\hskip
2.84544pt\theta_{2}={\int\limits}_{L_{1}}^{s}\dfrac{1}{\rho_{2}(s)}ds,\hskip
2.84544pt\tilde{\theta}_{1}={\int\limits}_{0}^{s}\theta_{1}ds,\hskip
2.84544pt\tilde{\theta}_{2}={\int\limits}_{L_{1}}^{s}\theta_{2}ds\;\;,$ (12)

The bending angle of the half dipole is then given by:

$\theta=\theta_{{1}(s=L_{1})}+\theta_{{2}(s=L_{1}+L_{2})}={\int\limits}_{0}^{L_{1}}\dfrac{1}{\rho_{1}(s)}ds+{\int\limits}_{L_{1}}^{L_{1}+L_{2}}\dfrac{1}{\rho_{2}(s)}ds\;\;.$
(13)

Inserting the partial dispersion invariants into Eq. (1), the emittance is
found to be:

$\epsilon_{x}=G\left(\dfrac{1}{L_{1}}{\int\limits}_{0}^{L_{1}}\dfrac{{\cal
H}_{1}}{\left|\rho_{1}(s)\right|^{3}}ds+\dfrac{1}{L_{2}}{\int\limits}_{L_{1}}^{L_{1}+L_{2}}\dfrac{{\cal
H}_{2}}{\left|\rho_{2}(s)\right|^{3}}ds\right),~{}\mathrm{where}~{}~{}G=\dfrac{C_{q}\gamma^{2}}{J_{x}}\left(\dfrac{1}{L_{1}}{\int\limits}_{0}^{L_{1}}\dfrac{1}{\rho_{1}(s)^{2}}ds+\dfrac{1}{L_{2}}{\int\limits}_{L_{1}}^{L_{1}+L_{2}}\dfrac{1}{\rho_{2}(s)^{2}}ds\right)^{-1}.$
(14)

The final form of Eq. (14) can be expressed as:

$\epsilon_{x}=G\dfrac{I_{7}+I_{8}\lambda+(I_{1}+I_{2}\lambda)\beta_{cd}^{2}+\eta_{cd}\left(I_{5}+I_{6}\lambda+(I_{3}+I_{4}\lambda)\eta_{cd}\right)}{L_{1}\beta_{cd}}\;\;,$
(15)

with the integrals $I_{1}-I_{8}$ given in Appendix B.

In order to find the absolute minimum emittance, the first partial derivatives
of the emittance with respect to the beta and dispersion functions should be
zeroed. As a result, the $\beta_{TME}$ and $\eta_{TME}$ values that achieve
the TME at the center of the dipole are found to be:

$\beta_{TME}=\dfrac{\sqrt{-(I_{5}+I_{6}\lambda)^{2}+4(I_{7}+I_{8}\lambda)(I_{3}+I_{4}\lambda)}}{2\sqrt{(I_{1}+I_{2}\lambda)(I_{3}+I_{4}\lambda)}}\hskip
5.69046pt\mathrm{and}\hskip
5.69046pt\eta_{TME}=-\dfrac{I_{5}+I_{6}\lambda}{2(I_{3}+I_{4}\lambda)}\;\;.$
(16)

By inserting Eq. (16) into Eq. (15) the expression for the TME is given by:

$\epsilon_{TME}=G\dfrac{\sqrt{I_{1}+I_{2}\lambda}\sqrt{-(I_{5}+I_{6}\lambda)^{2}+4(I_{7}+I_{8}\lambda)(I_{3}+I_{4}\lambda)}}{L_{1}\sqrt{I_{3}+I_{4}\lambda}}~{}.$
(17)

The emittance detuning factor $\epsilon_{r}$ that describes the deviation of
the emittance $\epsilon_{x}$ from its theoretical minimum $\epsilon_{TME}$ is
given by:

$\epsilon_{r}=\dfrac{\epsilon_{x}}{\epsilon_{TME}}=\dfrac{\sqrt{I_{3}+I_{4}\lambda}\left(I_{7}+I_{8}\lambda+(I_{1}+I_{2}\lambda)\beta_{cd}^{2}+\left(I_{3}+I_{4}\lambda\right)\eta_{cd}^{2}-\left(I_{5}+I_{6}\lambda\right)\eta_{cd}\right)}{\beta_{cd}\sqrt{I_{1}+I_{2}\lambda}\sqrt{-(I_{5}+I_{6}\lambda)^{2}+4(I_{7}+I_{8}\lambda)(I_{3}+I_{4}\lambda)}}$
(18)

Solving Eq. (18) with respect to $\beta_{cd}$ gives:

$\beta_{cd_{1,2}}=\dfrac{\epsilon_{r}\sqrt{4\left(I_{7}+I_{8}\lambda\right)-\dfrac{\left(I_{5}+I_{6}\lambda\right)^{2}}{I_{3}+I_{4}\lambda}}\pm\sqrt{4\left(\left(I_{7}+I_{8}\lambda\right)\left(\epsilon_{r}^{2}-1\right)+\left(I_{3}+I_{4}\lambda\right)\eta_{cd}^{2}-\left(I_{5}+I_{6}\lambda\right)\eta_{cd}\right)-\dfrac{\left(I_{5}+I_{6}\lambda\right)^{2}}{I_{3}+I_{4}\lambda}\epsilon_{r}^{2}}}{2\sqrt{I_{1}+I_{2}\lambda}}\;\;.$
(19)

Applying the requirement of $\beta_{cd}$ to be a real-positive number, the
quadratic dependence of the argument in the square root on the dispersion at
the center of the dipole must have an upper and a lower limit, i.e.
$\eta_{min}<\eta_{cd}<\eta_{max}$, given by:

$\eta_{cd_{min,max}}=-\dfrac{(I_{5}+I_{6}\lambda)\pm\sqrt{((I_{5}+I_{6}\lambda)^{2}-4(I_{3}+I_{4}\lambda)(I_{7}+I_{8}\lambda))(1-\epsilon_{r}^{2})}}{2(I_{3}+I_{4}\lambda)}\;\;.$
(20)

The $\beta_{cd}$ has two solutions for a fixed $\eta_{cd}$. The solutions of
$\beta_{cd}$ and $\eta_{cd}$, that depend on the detuning factor
$\epsilon_{r}$, determine the limits of $\epsilon_{x}$.

The horizontal and vertical phase advances of the cell can be found using the
trace of the cell transfer matrix. The horizontal phase advance, using Eq.
(7), is given by:

$\displaystyle\cos\mu_{x}=$
$\displaystyle\dfrac{g_{2}^{2}-\beta_{cd}^{2}\theta^{2}}{g_{2}^{2}+\beta_{cd}^{2}\theta^{2}}\;\;.$
(21)

For a uniform dipole, the horizontal phase advance in order to reach the
absolute minimum emittance is independent of any cell or dipole
characteristics and has the unique value $\mu_{x}=284.5^{\circ}$ Streun
(2006). However, in the case of the non-uniform dipoles, the horizontal phase
advance for reaching the TME condition depends on $\theta$ and
$\tilde{\theta}$ and thereby, on $\tilde{\rho}$ and $\lambda$. The vertical
phase advance can be expressed as:

$\begin{split}\cos\mu_{y}=&1+(L+2(s_{13}+s_{12}))(\frac{1}{f_{1}}+\frac{1}{f_{2}})+2\frac{(L+s_{1}+s_{12})s_{23}+s_{12}s_{3}}{f_{1}f_{2}}+\frac{(L+2s_{1})s_{23}}{f_{1}^{2}}\\\
&+\frac{(L+2(s_{1}+s_{12}))s_{3}}{f_{2}^{2}}+\frac{(L+2s_{1})(s_{2}^{2}+2s_{2}s_{3})}{f_{1}^{2}f_{2}}+2\frac{(L+2s_{1}+s_{12})s_{2}s_{3}}{f_{1}f_{2}^{2}}+\frac{(L+2s_{1})s_{2}^{2}s_{3}}{f_{1}^{2}f_{2}^{2}}\;\;,\end{split}$
(22)

where $s_{12}=s_{1}+s_{2}$, $s_{13}=s_{1}+s_{3}$ and $s_{23}=s_{2}+s_{3}$. If
the cell is tuned to the absolute minimum emittance conditions, for $s_{1}\to
0$ or $s_{2}\to 0$ or $s_{3}\to 0$ and based on the results presented in
Appendix A, the $\cos\mu_{y}$ goes to infinity and so, the vertical motion is
unstable. Unlike the horizontal phase advance, the vertical one depends both
on the optics functions at the dipole center and on the cell geometry.

##  III DIPOLE PROFILES

Based on studies of preceding works for the longitudinally variable bends Guo
and Raubenheimer (2002); Nagaoka and Wrulich (2007); Papaphilippou and
Elleaume (2005); Papaphilippou  _et al._ (2005); Wang  _et al._ (2011); Wang
(2009); Papadopoulou ; Streun and Wrulich (2015); Papadopoulou  _et al._
(2015), two dipole profiles are presented where the bending radius forms a
step and a trapezoidal shape. The step profile shown in Fig. 4 (left) consists
of two different constant field segments, having the minimum bending radius at
the dipole center. The evolution of $\rho$ for the step profile is given by:
$\rho(s)=\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}\rho_{1},~{}0<s<L_{1}\\\
\rho_{2},~{}L_{1}<s<L_{1}+L_{2}=L/2\end{array}\end{array}$
The trapezium profile is shown in Figure 4 (right), where again the strongest
constant field segment is localized at the center of the dipole. The evolution
of the bending radius from the dipole center until its edge is expressed as:
$\rho(s)=$
$\begin{array}[]{cc}\Bigg{\\{}&\begin{array}[]{l}\rho_{1},~{}0<s<L_{1}\\\
\rho_{1}+(L_{1}-s)(\rho_{1}-\rho_{2})/L_{2},~{}L_{1}<s<L_{1}+L_{2}=L/2\end{array}\end{array}$

Figure 4: The evolution of the beding radius along the step (left) and the
trapezium (right) dipole profile.

The theoretical minimum emittance, as calculated using Eq. (17) for each
dipole profile, depends on $\tilde{\rho}$, $\lambda$ and $\theta$. The
reduction factor that describes the reduction of the minimum emittance for a
non-uniform dipole with respect to a uniform one with the same bending angle,
is defined as:

$F_{TME}=\dfrac{\epsilon_{TME_{uni}}}{\epsilon_{TME_{var}}}\;\;,$ (23)

where $\epsilon_{TME_{uni}}$ and $\epsilon_{TME_{var}}$ are the TMEs for a
uniform dipole and a variable bend respectively. The $F_{TME}$ depends only on
$\tilde{\rho}$ and $\lambda$. For both dipole cases, the full expression of
the $F_{TME}$ is given in Appendix C. In practice, the TME cells are detuned
to reach larger emittances so that the cell characteristics are more relaxed.
Moving away from the TME, the resulted emittances are $\epsilon_{var}$ and
$\epsilon_{uni}$ for the non-uniform and for the iso-magnetic dipoles,
respectively. In order to compare the emittances of a uniform and of a non-
uniform bending magnet, their ratio (using Eqs. (18) and (23)) is defined as:

$\dfrac{\epsilon_{var}}{\epsilon_{uni}}=\dfrac{\epsilon_{r_{var}}\epsilon_{TME_{var}}}{\epsilon_{r_{uni}}\epsilon_{TME_{uni}}}=\dfrac{\epsilon_{r_{var}}}{\epsilon_{r_{uni}}}\dfrac{1}{F_{TME}}\;\;,$
(24)

where $\epsilon_{r_{var}}$ and $\epsilon_{r_{uni}}$ are the detuning factors
for the non-uniform and for the uniform dipole respectively. In order to get
an emittance reduction, it should always be
$\dfrac{\epsilon_{var}}{\epsilon_{uni}}<1$. Thus, using Eq. (24), the
restriction of:

$\dfrac{\epsilon_{r_{var}}}{\epsilon_{r_{uni}}}<F_{TME}$ (25)

is established. The smaller is the ratio of the detuning factors compared to
the $F_{TME}$ (that is fixed in accordance to the chosen dipole
characteristics), the lower is the final emittance the variable bend gives.
Practically this means that even if the detuning of a TME cell with a variable
bend is larger than in the case of using a uniform dipole, emittance
reductions are possible if Eq. (25) is satisfied.

The characteristics of a realistic dipole profile are driven by the constraint
of how sharply and quickly the transition from the high to the low field can
be established. Regarding the fact that the fringe field of the first dipole
part should not significantly affect the field of the second one and that a
sharp field drop off is technologically questionable, the minimum difference
between the highest and the lowest field and the corresponding difference in
the lengths is assumed to be $\lambda,\tilde{\rho}>0.04$. Based on the optimal
variable bend characteristics, a design study is on-going in order to provide
the final specifications of the dipole to be fabricated Domínguez Martinez and
Toral ; Domínguez Martinez  _et al._ (2018).

Figure 5: The parametrization of the reduction factor $F_{TME}$ with the ratio
of the bending radii $\tilde{\rho}$ and lengths $\lambda$ for the step (left)
and the trapezoidal (right) profile, for the TME case. The black contour lines
correspond to different values of horizontal phase advances.

In Figure 5, the reduction factor $F_{TME}$ is parametrized with
$\tilde{\rho}$ and $\lambda$, for the step (left) and the trapezoidal (right)
profile. The areas where $F_{TME}$ is high are blue-colored, while red-colored
are the areas where the reduction is smaller. The black contour lines show
different values of the horizontal phase advance. As mentioned above, for a
uniform dipole, in order to reach the absolute minimum emittance, there is a
unique horizontal phase advance of $\mu_{x}=284.5^{\circ}$ Streun (2006),
independent of any cell or dipole characteristics. However, in the case of the
non-uniform dipoles, the horizontal phase advance for reaching the TME
condition depends on $\tilde{\rho}$ and $\lambda$. The highest reductions
correspond to high phase advances $\mu_{x}>310^{\circ}$. Remarkable emittance
reductions are reached even for lower phase advances which correspond to
smaller chromaticities.

For both profiles, in the limits where $\tilde{\rho}$, $\lambda\rightarrow 1$
(i.e. $\rho_{2}=\rho_{1}$ and $\lambda_{2}=\lambda_{1}$) there is no emittance
reduction. In the limits where $\lambda\rightarrow 0$ and
$\tilde{\rho}\rightarrow 0$ (i.e. $L_{2}>>L_{1}$ and $\rho_{2}>>\rho_{1}$) the
reductions obtained are practically infinite. The highest possible reductions
are found to be around 13 and 34 for the step and the trapezium profile,
respectively, for $\lambda,\tilde{\rho}>0.04$. These reductions are localized
where both $\lambda$ and $\tilde{\rho}$ are low, demanding a sharp transition
from the high to the low field region. The issue of concern for the design of
a variable bend is how small can $\tilde{\rho}$ be in order to get a realistic
difference between the maximum and the minimum magnetic field along a specific
dipole length that has a fixed bending angle.

In order to facilitate the comparison between the step and the trapezium
profile, the number of dipoles $N_{d}$, their length $L$ and the minimum
bending radius $\rho_{1}$ values are kept the same. As a numerical example,
the minimum $\rho_{1}$ value is set to $4.1$ m, i.e. $B=2.3$ T at an energy of
$2.86$ GeV for the CLIC DRs. In addition, examples for dipole lengths and
angles different than the ones of the CLIC DR design are presented. Using Eq.
(13), the bending angles for the step and the trapezium profile respectively,
are found to be:

$\theta_{step}=\dfrac{L(\lambda+\tilde{\rho})}{\rho_{1}(1+\lambda)}~{}~{}~{}\mathrm{and}~{}~{}~{}\theta_{trapezium}=\dfrac{L(\lambda(-1+\tilde{\rho})+\tilde{\rho}\ln\tilde{\rho})}{\rho_{1}(-1+\tilde{\rho})(1+\lambda)}\;\;.$
(26)

Solving Eq. (26) with respect to the ratio $\tilde{\rho}$, for a fixed bending
radius $\rho_{1}$, provides an expression of $\tilde{\rho}$ depending on the
bending angle $\theta$, the dipole length $L$ and the lengths ratio $\lambda$.
Thereby, the reduction factor of Eq. (23) becomes a function of the bending
angle $\theta$ (or the number of bends $N_{d}$), the dipole length $L$ and the
lengths ratio $\lambda$. Even if the reduction factors achieved get higher by
increasing the number of dipoles and their length, a compromise between these
two parameters is required. Additionally, as shown in the next section, the
fabrication of a variable bend sets a lower limit on the length ratio
$\lambda$ Domínguez Martinez  _et al._ (2018) and thus, an upper limit on the
reduction factor values that can be achieved.

Figure 6: The parametrization of the $F_{TME}$ with $\tilde{\rho}$ and
$\lambda$, when fixing $N_{d}$ and $L$, for the step (left) and the trapezium
(right) profile.

After imposing the bending angle $\theta$, the dipole length $L$ and minimum
bending radius $\rho_{1}$ in Eq. (26), a relationship between the ratios
$\tilde{\rho}$ and $\lambda$ is obtained and can be inserted in Eq. (23) for
obtaining the respective emittance reduction factors. In this way, the maximum
emittance reductions for fixed dipole characteristics can be found. Figure 6
shows the parameterization of the reduction factor $F_{TME}$ with the ratios
$\tilde{\rho}$ and $\lambda$, for both dipole profiles and again with the
restriction of $\lambda,\tilde{\rho}>0.04$, for three different cases of
dipole numbers and lengths ($N_{d}$, $L$) equal to ($100$, $0.7$ m), ($100$,
$0.58$ m) and ($90$, $0.58$ m). The case corresponding to $100$ dipoles with a
length of $0.58$ m matches the exact dipole characteristics of the CLIC DRs.
For the dipole constraints applied in each case, the trapezium gives always
higher reductions than the step profile. Obviously, the more and the longer
the dipoles are, the higher emittance reductions are achieved. The maximum
reductions in all cases are localized where both $\lambda$ and $\tilde{\rho}$
are low. Large length ratios are not of interest since the reduction factor
becomes very small. The bending radius ratios resulting in the highest
emittance reductions are lower for the trapezoidal profile. The variation of
the bending radius along the dipole length that results in the highest
emittance reduction for each case, is plotted in Fig. 7 for the trapezium
profile. As was expected from Fig. 6, the maximum emittance reductions (where
$\lambda=0.04$) correspond to different ratios $\tilde{\rho}$ depending on the
number of dipoles and length and thus, for a fixed minimum bending radius
($\rho_{1}$=$4.1$ m), their maximum bending radius differs.

Figure 7: The variation of the bending radius along the dipole length for
different ($N_{d}$, $L$) pairs, resulting in the highest emittance reduction
(where $\lambda=0.04$) for each case, for the case of the trapezium profile.

##  IV Longitudinally variable field dipole design for the CLIC Damping Rings

The analytical results showed that the trapezium dipole profile can reach very
low emittances, compared to a uniform dipole of the same bending angle.
Therefore, the fabrication of a variable bend having a bending radius that
forms a trapezium shape is of interest. According to the optimal
characteristics of a trapezium dipole profile to be used for the CLIC Damping
Rings, the magnetic design of a longitudinally variable bending dipole based
on permanent magnets was studied and the prototype will be fabricated by
CIEMAT Domínguez Martinez and Toral ; Domínguez Martinez  _et al._ (2018).

Figure 8: Magnet design based on the characteristics of the variable bends for
the CLIC DRs Domínguez Martinez and Toral ; Domínguez Martinez  _et al._
(2018).

The main challenges of this design are the bending radius variation which
should change linearly along the magnet and the high field region length that
is very short. The longitudinal gradient with a trapezoid decay is solved by
splitting the magnet in three differentiated field regions combined with an
innovative variable gap solution, as presented in Fig. 8.

Figure 9: The field variation along a dipole having a peak field of 2.3 T; the
designed trapezium profile (red colored), the resulted from analytical
calculations profile (black colored) and the simulated in MADX field profile
(blue colored).

The low field block is made of SmCo magnets. The medium field has the same
configuration as the high field section, using NdFeB blocks. The requested
peak field was initially limited to 1.77 T as a reasonable value for a non-
superconducting magnet, requiring to deal with iron saturation that is
partially overcome using a Fe-Co material and suppressing the hyperbolic
profile in the high field region pole tip which is the most saturated section.

The higher is the magnetic field at the center of a variable bend, the higher
emittance reduction factors can be reached. The use of permanent magnets could
allow having a higher field than in the case of a normal conducting magnet.
The 3D simulations performed have shown that the peak field could be increased
above 2 T, resulting in higher emittance reduction factors. In Fig. 9, a 2.3 T
designed trapezium profile is shown in red. The field decay successfully
matches the desired from the hyperbolic field profile provided by the
analytical estimates (black colored). For the designed trapezium profile, the
$\lambda$ and $\tilde{\rho}$ values reached are respectively 0.036 and 0.295,
corresponding to an emittance reduction factor of $F_{TME}=7.1$ 222The
technological restrictions do not allow $\tilde{\rho}$ to reach the optimal
value of 0.263 for a $\lambda=0.036$, providing a reduction factor
$F_{TME}=8.3$.. For the beam optics simulations performed with MADX MAD-X
homepage , the field of the designed trapezium profile is being approximated
by a sequence of dipoles with a step-like field (blue colored). Finally, since
a transverse gradient of -11 T/m was requested, the designed magnet provides
at the same time dipolar and quadrupolar fields, having pole tips with a
hyperbolic profile to produce the gradient.

##  V Numerical application for a VARIABLE BEND TME CELL

In this section, a numerical application of the aforementioned analytical
results are presented for the CLIC DR TME cell. The DRs provide the final
stage of damping of the $e^{+}e^{{-}}$ beams of the linear collider after the
pre-damping rings and their lattice has a racetrack shape with two arc
sections and two long straight sections (LSS), with the TME cells being placed
in the arcs Papaphilippou  _et al._ (2012). The required output horizontal
normalized emittance for the CLIC DR, at the energy of $2.86$ GeV, is $500$
nm-rad, i.e. around 90 pm geometrical emittance. According to the design of a
variable bend, the maximum dipole field is set to be $2.3$ T (corresponding to
the minimum bending radius $\rho_{1}=4.1$ m), for dipoles having a length of
$L=0.58$ m. The maximum pole tip field of the quadrupoles and the sextupoles
is $B_{q}^{\mathrm{max}}=1.1$ T and $B_{s}^{\mathrm{max}}=0.8$ T respectively.
By fixing these quantities, the free parameters left are the drift space
lengths $s_{1}$, $s_{2}$, $s_{3}$ and the emittance, based on optics stability
and feasibility constraints described below. It should be stressed that, it is
always preferable to carefully detune the cell from the TME to larger
emittances (see Eq. (25)), so that the optics characteristics of the cell and
in particular chromaticity is under control.

###  V.1 Stability and feasibility constraints

The optics stability criterion, for both horizontal and vertical planes is
given by:

$|cos\mu_{x,y}|<1\;\;.$ (27)

The pole tip field value of both quadrupoles and sextupoles has a limit and
the radius of the magnet aperture has a minimum value. The feasibility of the
quadrupoles is ensured if the quadrupole strength $k$ is kept below a maximum
value given by:

$\displaystyle
k=\dfrac{1}{fl_{q}}=\leq\dfrac{1}{(B\rho)}\dfrac{B_{q}^{\mathrm{max}}}{R_{\mathrm{min}}}\;\;,$
(28)

where $B\rho$ is the magnetic rigidity and $B_{q}^{\mathrm{max}}$ is the
quadrupole pole tip field. The
$R_{\mathrm{min}}=\sqrt{\beta\epsilon_{\mathrm{max}}+((\frac{\delta
p}{p_{0}})_{\mathrm{max}}D)^{2}}$ is the minimum required aperture radius for
a Gaussian beam distribution, where $\epsilon_{\mathrm{max}}$ is the maximum
value of the Courant-Snyder invariant for the injected beam and $(\delta
p/p_{0})_{\mathrm{max}}$ the corresponding total momentum deviation. As the
sextupoles are set to cancel the chromaticity induced by the quadrupoles,
their strength can be calculated by
$\xi_{x,y}=-\frac{1}{4\pi}\oint{\beta_{x,y}[K_{x,y}(s)-S(s)D(s)]ds}=0$, where
$K_{x,y}$ correspond to the focusing and defocussing quadrupole strengths.
Denoting the pole tip field of the sextupoles as $B_{s}^{\mathrm{max}}$, their
feasibility is ensured if the strength $S$ is lower than a maximum value:

$\displaystyle
S\leq\dfrac{2B_{s}^{\mathrm{max}}}{R_{\mathrm{min}}^{2}}\dfrac{1}{(B\rho)}\;\;.$
(29)

###  V.2 Parametrization with the drift lengths

The dependence of different cell characteristics on the drift space lengths
require their parametrization with $s_{1}$, $s_{2}$, $s_{3}$. A scanning of
drift space lengths was performed, aiming to solutions with low chromaticities
and small quadrupole strengths, while keeping the cell compact. The
chromaticities were calculated for all combinations of drift lengths when
$s_{1}[m]\in[0.1,2]$, $s_{2}[m]\in[0.1,2]$ and $s_{3}[m]\in[0.1,1]$ and the
results are presented in Fig. 10. In these plots, the drift lengths are color-
coded with the horizontal (left) and vertical (right) chromaticity.

Figure 10: The horizontal (left) and vertical (right) chromaticities as
parametrized with the drift lengths $s_{1},s_{2},s_{3}$ (top) for the TME with
the trapezium dipole profile. In the bottom plot, the same parametrization is
plotted and projected to the $s_{1},s_{2}$ plane. High absolute chromaticities
correspond to red, whereas low absolute values are blue.

The lowest absolute chromaticities in both planes are found for $s_{1}<0.5$ m.
The absolute minimum emittances exist only for large negative chromaticities
per cell, i.e. for $\xi_{x}<-2$ and $\xi_{y}<-1$. Detuning factors that give
emittances larger than the TME can provide solutions with lower negative
chromaticities. Although a careful choice of the first and second drift
lengths ($s_{1}$ and $s_{2}$) is essential, the cell parameters are less
dependent on the third drift length $s_{3}$. In this respect, in order to
clearly distinguish the two regions of optically stable solutions, the
parametrization of the chromaticities as projected in the $s_{1},s_{2}$ plane
is presented in Fig. 10 (bottom). The horizontal chromaticity depends strongly
on drift length between the two quadrupoles $s_{2}$. In fact, an area of low
horizontal chromaticities can be found, whereas low vertical chromaticities
can be obtained following a thin curve. On the other hand, only a small
fraction of the ($s_{1}$, $s_{2}$) combinations satisfy the requirements of
having focal lengths leading to feasible magnets. In this respect, the optimal
values indicated with a magenta colored point are found to be $s_{1}=0.33$ m,
$s_{2}=0.28$ m and $s_{3}=0.18$ m, slightly on the right side of the lowest
chromaticity areas.

###  V.3 Parameterization with the emittance

After fixing the drift lengths, cell properties such as the quadrupole
strength can be parametrized with the emittance detuning factor. The emittance
value that was so far determined by the reduction factor $F_{TME}$, is
increased with this detuning. The emittance reduction due to the non-uniform
dipole profile as compared to a standard dipole depends on the relation
between the detuning and the reduction factor (see Eq. (24)). Using this
relation, the additional benefit for using longitudinally variable bends
instead of the standard ones can be clearly demonstrated. The plots in Fig. 11
provide the parametrization of the horizontal optics functions at the center
of the dipole and the quadrupole strengths color-coded with the detuning
factor. Black points correspond to solutions that satisfy the optics stability
in both planes. The magenta points are the ones satisfying low chromaticities
and quadrupole strengths.

Figure 11: Parameterization of the beta and dispersion functions at the dipole
center $\beta_{cd}$, $\eta_{cd}$ (left) and of the focal lengths, for
$f_{1}>0$, $f_{2}<0$ (middle) and for $f_{1}<0$, $f_{2}>0$ (right), with the
detuning factor. The black squares indicate stability and the magenta ones
feasibility-low chromaticity solutions.

The ellipsoidal curves in Fig. 11 (left) represent the pairs of beta and
dispersion functions at the dipole center which result in the same emittance
(see Eq. (18)). Similarly, the parametrization of the focal lengths with the
emittance detuning factor is given in Fig. 11 (middle) for $f_{1}>0$ and
$f_{2}<0$, where the pairs of ($f_{1}$, $f_{2}$) lie again on constant
emittance curves. Solutions with $f_{1}<0$ and $f_{2}>0$ which correspond to
the modified TME cell Jiao  _et al._ (2011), also exist and they are presented
in Fig. 11 (right). The TME ($\epsilon_{r}=1$) is achieved for a unique pair
of beta and dispersion functions that is ($\eta_{cd},\beta_{cd})=(1.1\times
10^{-4},0.07$) m and only for one pair of focal lengths which is
($f_{1},f_{2})=(0.27,-0.18)$ m (out of scale in the bottom right corner of the
right part of Fig. 11), i.e. quite strong quadrupole strengths are necessary.
Furthermore, and for the chosen drift lengths, there are no solutions that
assure optics stability and low chromaticities. Solutions that assure lattice
stability (black points) and for chromaticities $\xi_{x},~{}\xi_{y}>-2.5$
(magenta points) arise when moving away from the TME to larger detuning
factors. Solutions with both focal lengths positive are unstable. Even if the
chosen cell characteristics result in a confined $\epsilon_{r}$ region, the
low emittances reached for a very compact cell counteracts the fact that it is
numerically challenging to tune the cell. One can finally observe that there
is a well-determined focal length range in the middle plot of Fig. 11, for
which the cell can be tuned, satisfying stability and feasibility criteria.
For the opposite quadrupole polarity (first quadrupole de-focusing, see right
plot of Fig. 11), the zone spans a larger focal length range, confined,
though, to a narrow line, which may result in a difficulty to reach
numerically the desired solution.

(a)

(b)

(c)

Figure 12: Parameterization of the detuning factor (a) and the chromaticities
$\xi_{x}$ (b) and $\xi_{y}$ (c) with the horizontal $\mu_{x}$ and vertical
$\mu_{y}$ phase advances, for $f_{1}>0$, $f_{2}<0$, for the trapezium dipole
profile.

Figure 12 shows the parametrization of the detuning factor $\epsilon_{r}$ (a)
and of the horizontal $\xi_{x}$ (b) and vertical $\xi_{y}$ (c) chromaticity
with the horizontal $\mu_{x}$ and vertical $\mu_{y}$ phase advances, for the
case of $f_{1}>0$, $f_{2}<0$ solutions which appear only when
$\mu_{x}>0.5\cdot 2\pi$. Towards high vertical phase advances, the
chromaticities for both planes have high negative values ($\xi_{x}$,
$\xi_{y}<-3$). Large horizontal phase advances correspond to minimum
dispersion and beta functions at the center of the dipole that require strong
focusing and to that end, result in high chromaticities. It can be noticed
that for $\mu_{y}<0.5\cdot 2\pi$, there are low negative chromaticities even
for small detuning factors corresponding to emittances close to the minimum
one.

(a)

(b)

(c)

Figure 13: Parameterization of the detuning factor (a) and the chromaticities
$\xi_{x}$ (b) and $\xi_{y}$ (c) with the horizontal $\mu_{x}$ and vertical
$\mu_{y}$ phase advances, for modified TME cell where $f_{1}<0$, $f_{2}>0$,
for the trapezium dipole profile.

Solutions for $f_{1}<0$, $f_{2}>0$, corresponding to the modified TME cell
Jiao  _et al._ (2011), also satisfy the stability criteria for the chosen cell
characteristics and are presented in Fig. 13. These solutions appear always
for $\mu_{x}<0.5\cdot 2\pi$. The quadrupole strengths for the modified TME
cell are lower compared to the normal TME cell, as presented in Fig. 11
(right). Knowing the $F_{TME}$ for the designed variable bend, the restriction
described in Eq. (25) sets an upper limit to the detuning of the cell. Since
the modified TME cell requires a large detuning in order to get low horizontal
chromaticities, the final emittance reductions reached are not sufficient. In
this respect, the normal TME cell (having $f_{1}>0$, $f_{2}<0$ solutions) will
be further used for the following numerical application to the CLIC DR optics
design.

##  VI Alternative CLIC DR design based on longitudinally variable bends and
high field wigglers

As discussed in the previous section, the DR lattice Papaphilippou  _et al._
(2012) has a racetrack shape with arc sections composed by TME cells. The two
long straight sections (LSS) are composed by FODO cells housing RF cavities,
injection and extraction equipment and mainly damping wigglers. The DR lattice
design is driven by the emittance requirements which for ultra-low values give
rise to collective effects Papaphilippou  _et al._ (2012), with intra-beam
scattering (IBS) being the dominant one. An alternative design is proposed,
which aims to mitigate the IBS effect for a compact ring, using in the arc
cells the designed variable bend presented earlier and an optimized high-field
wiggler in the FODO cells. The optimization steps followed, as well as the
final parameters for the improved design are discussed in this section.

###  VI.1 Optimization of the arc TME cell

For the beam optics simulations performed with the MADX code MAD-X homepage ,
the field variation of the designed trapezium profile is being approximated by
a sequence of dipoles with varying field, as illustrated in Fig. 9. The dipole
length is $L=0.58$ m and the maximum dipole field is $2.3$ T. When the uniform
dipoles of the current design are replaced by variable bends, the resulted
emittance is lower than the required one. In this case, the subtraction of
some TME cells from the existing arc is possible. Actually, the number of
dipoles (i.e. total number of TME cells) can be reduced to such an extent that
the required emittance is still achieved, thereby resulting in a shorter ring.

As soon as the characteristics of the dipole are fixed, the drift space
lengths are chosen in accordance to the results presented in section IV. For a
combined function dipole, i.e. having a small defocussing gradient (see
section IV), the IBS effect is reduced through the increase of the vertical
beam size at the center of the bend. Basically, instead of having a low
$\beta$ at the center of the dipole in both planes, the optics matching
imposes $\beta_{y}$ to be maximum there. Therefore, there is a reduction of
IBS growth rates. Keeping in mind that a TME cell with a combined function
dipole reduces the IBS effect Braun  _et al._ (2010), assists in choosing the
proper phase advances, that guarantee low chromaticities and small quadrupole
strengths. The parametrization with the emittance has shown that the
quadrupole strengths for which $f_{1}>0$ and $f_{2}<0$ (Fig. 12), can only be
found for $\mu_{x}>0.5\cdot 2\pi$. A good compromise for the horizontal phase
advance is to be around $0.51\cdot 2\pi$ and for the vertical phase advances
to be always below $0.5\cdot 2\pi$. After a detailed scanning of the cell
characteristics, the horizontal and vertical phases advances of the TME cell
are respectively chosen to be around $0.51\cdot 2\pi$ and $0.11\cdot 2\pi$.
Taking into account the emittance reduction and detuning factors, emittances
smaller than the ones of the current design with the uniform dipoles, are
reached. In this respect, it was possible to reduce the number of dipoles down
to $N_{d}=90$ for the case of the designed trapezium profile with 2.3 T
maximum field.

###  VI.2 Optimization of the FODO cell

Figure 14: The dependence of the steady state emittances ($\epsilon_{xIBS}$
and $\epsilon_{yIBS}$) color-coded with their ratio with respect to the
corresponding equilibrium emittances ($\epsilon_{x0}$ and $\epsilon_{y0}$) on
the wiggler peak field $B_{w}$, for the trapezium dipole profile.

Almost every FODO structure of the LSS accommodates two wigglers. The ultimate
purpose of using damping wigglers is to further reduce damping times and
thereby maintain low emittances by reducing the impact of IBS, but also of
various collective effects. At the same time, the fast damping time is imposed
by the 50 Hz repetition rate of the collider. The use of super-conducting
technology is mandatory in order to have a high wiggler field and a relatively
short period for obtaining low emittances and fast damping time. The super-
conducting magnet wigglers used in the current design have a $B_{w}=2.5$ T
peak field and $\lambda_{w}=5$ cm period.

It was shown that by targeting higher wiggler fields not only the emittance
but also the IBS effect can be reduced Antoniou (2012); Schoerling  _et al._
(2012). Taking into account the optimization of the arc cells and the fact
that the emittance with IBS is significantly lowered after increasing the
wiggler’s peak field, the FODO cells per LSS can be reduced from 13 down to
10. The plots in Fig. 14 show the MADX results of the parametrization of the
steady state transverse emittances including the IBS effect with the wiggler
peak field $B_{w}$, starting from the 2.5 T that is the field of the existing
wiggler design, for 10 FODO cells. Clearly, the wiggler field increase
corresponds to a significant reduction of the IBS effect. The new working
point for the damping wigglers, that complies with the technological
restrictions, is proposed to be at 3.5 T and with a 49 mm period length,
giving an output emittance of around 435 nm-rad. This design necessitates a
different wire technology, using Nb3Sn material García Fajardo  _et al._
(2016).

###  VI.3 Optical functions and new design parameters

Figure 15: Optical functions of the TME cell (left) and the dispersion
suppressor-beta matching section followed by the FODO cell (right), when using
in the arcs the trapezium dipole profiles.

The final lattice, with a smaller number of dipoles and wigglers than the ones
of the existing design, is produced. In Fig. 15 (left), the matched optics,
i.e. horizontal dispersion, horizontal and vertical beta functions, are
plotted for one arc TME cell. On the top part of the figure, a schematic
layout of the cell is presented, showing the two doublets of quadrupoles and
the sextupoles that are placed between the two mirror symmetric defocussing
quadrupoles and between the dipole and the focusing quadrupoles. In Fig. 15
(right), the matched optics of the dispersion suppressor-beta matching section
followed by the wiggler FODO cell, are presented.

The parameters of the original design and the alternative ones are displayed
in Table 1. The alternative designs have 90 dipoles and 40 FODO cells with
wigglers of $3.5$ T peak field, while the original has 100 uniform dipoles and
52 FODO cells with wigglers of $2.5$ T peak field. The parameters for the
alternative new design are given for the case of uniform and variable bends,
keeping the same arc TME cells and FODO cells, so as to have a fair
comparison. By having exactly the same arc cells (length, angle and phase
advances), the emittance is reduced by $40~{}\%$ only due to the use of the
variable bends. For the alternative design, the optimal horizontal phase
advance is slightly above 180 deg., resulting in larger quadrupole strengths
whose length had to be increased in order to keep the gradient below $100$ T/m
and so, the total length of the TME cell became $7~{}\%$ longer. On the other
hand, the chromaticity is significantly smaller for the new DR design. Both
the original and the alternative design with variable bends can reach the
target emittances including IBS (as calculated by the Bjorken-Mtingwa
formalism through MADX MAD-X homepage ), with the alternative one giving a
$13~{}\%$ shorter ring circumference. Apart from achieving a lower emittance
using the trapezium profile, it is possible to allow significant margin for
the $500$ nm target, for an eventual increase of the required bunch
population, as lately proposed due to the CLIC re-baselining Boland  _et al._
(2016). Indeed, for the case of low energy CLIC which needs a higher bunch
charge of $N_{b}=5.7\times 10^{9}$ the final emittances are increased up to
$472$ nm and $4.6$ nm in the horizontal and vertical plane, respectively,
still remaining below the design target.

Table 1:  Design parameters for the original and the improved design of the CLIC DRs, for the case of $N_{b}=4.1\times 10^{9}$ Parameters, Symbol [Unit] | Original design | Alternative design
---|---|---
uniform | uniform | trapezium
Energy, $E$ [GeV] | 2.86 | 2.86
Bunch population, $N_{b}$ [$10^{9}$] | 4.07 | 4.07
Circumference, $C$ [m] | 427.5 | 373.7
Basic cell type in the arc/LSS | TME/FODO | TME/FODO
Number of arc cells/wigglers, $N_{d}/N_{w}$ | 100/52 | 90/40
RF Voltage, $V_{RF}$ [MV] | 4.50 | 6.50
Harmonic number, $h$ | 2850 | 2493
RF Stationary phase [o] | 62.3 | 58.9 | 63.0
Momentum compaction, $\alpha_{c}$ [$10^{-4}$] | 1.3 | 1.3 | 0.88
Damping times, ($\tau_{x},\tau_{y},\tau_{l}$) [ms] | (1.98, 2.05, 1.04) | (1.24, 1.28, 0.33) | (1.19, 1.23, 0.31)
Energy loss/turn, $U$ [MeV] | 4.0 | 5.6 | 5.8
Quadrupole gradient strengths, ($k_{1}$, $k_{2}$) [T/m] | (26, -53) | (66, -98) | (67, -98)
Phase advances per arc cell, $(\mu_{x},\mu_{y})$ [$360^{o}$] | 0.408/0.050 | 0.510/0.110
Horizontal and vertical tune, ($Q_{x}$, $Q_{y}$) | (48.35, 10.40) | (51.16, 14.56) | (51.18, 14.55)
Horizontal and vertical chromaticity, ($\xi_{x}$, $\xi_{y}$) | (-113, -82) | (-57, -70) | (-67, -75)
TME cell length, $L_{cell}$ [m] | 2.44 | 2.62
Dipole field, ($B_{min},B_{max}$) [T] | (0.97, 0.97) | (0.97, 0.97) | (0.62, 2.32)
Lengths and bending radii ratio, ($\lambda,\tilde{\rho}$) | (1, 1) | (1, 1) | (0.04, 0.26)
Normalized gradient in dipole [m-2 or T/m] | -1.1 or -10.5 | -1.1 or -10.5
Wiggler peak field, $B_{w}$ [T] | 2.5 | 3.5
Wiggler length, $L_{w}$ [m] | 2 | 2
Wiggler period, $\lambda_{w}$ [cm] | 5.0 | 4.9
| without IBS |
Normalized horiz. emittance, $\gamma\epsilon_{x}$ [nm-rad] | 312.2 | 574.1 | 350.3
Normalized vert. emittance, $\gamma\epsilon_{y}$ [nm-rad] | 3.3 | 3.3
Energy spread (rms), $\sigma_{\delta}$ [$\%$] | 0.11 | 0.15
Bunch length (rms), $\sigma_{s}$ [mm] | 1.4 | 1.6
Longitudinal emittance, $\epsilon_{l}$ [keVm] | 4.4 | 5.8
| with IBS |
Normalized horiz. emittance, $\gamma\epsilon_{x}$ [nm-rad] | 478.9 | 648.7 | 434.7
Normalized vert. emittance, $\gamma\epsilon_{y}$ [nm-rad] | 5.0 | 4.5 | 4.2
Energy spread (rms), $\sigma_{\delta}$ [$\%$] | 0.11 | 0.15
Bunch length (rms), $\sigma_{s}$ [mm] | 1.5 | 1.6
Longitudinal emittance, $\epsilon_{l}$ [keVm] | 4.7 | 5.8
IBS factors hor./ver./long. | 1.53/1.52/1.08 | 1.13/1.35/1.01 | 1.24/1.26/1.02

###  VI.4 Dynamic aperture

Figure 16: The on and off momentum dynamic aperture of the DR for the
trapezium dipole profile, for $\delta p=0$, $\delta p=0.5\%$ and $\delta
p=-0.5\%$.

The on- and off-momentum dynamic aperture (DA) of the ring was estimated for
particles tracked with the PTC module of MADX MAD-X homepage . Fig. 16 shows
the maximum initial positions of particles that survived over 1000 turns,
normalized to the horizontal and vertical beam sizes, at the point of
calculation ($\sigma_{x}=273.0$ $\mu m$ and $\sigma_{y}=40.2$ $\mu m$). This
simulation includes the effect of chromaticity sextupoles and magnets fringe
fields but no other additional imperfection such as misalignments or magnet
errors. The results for $\delta p=0$ are shown in blue, for $\delta p=0.5\%$
in red and for $\delta p=-0.5\%$ in yellow. The dynamic aperture achieved is
remarkable (almost 14 mm in the horizontal plane and 5 mm in the vertical
plane), allowing very comfortable on-axis injection. A working point
optimization, with simulations including misalignments, coupling and their
correction, the non-linear effect of wigglers and space charge tune-shift, was
further studied to fully quantify the non-linear performance of the new design
Ghasem  _et al._ (2016), which was found robust and adequate.

##  VII Conclusions

An analytical parametrization of the TME cell, based on linear optics
considerations and the thin lens approximation, has been derived for the case
of longitudinally variable bends. By choosing dipole profiles with
longitudinally varying magnetic fields, subject to certain constraints, the
emittances reached are lower as compared to the ones achieved by uniform
dipoles. Among the non-uniform dipoles studied, it is found that the one
having a bending radius forming a trapezium profile provides the largest
emittance reduction. The design of the hyperbolic field profile that uses
permanent magnets reaches 2.3 T at the highest field region and performs very
well in terms of emittance reduction Domínguez Martinez  _et al._ (2018). This
innovative design could be applied in any low emittance ring.

The phase advances that determine the properties of a TME cell with the
designed variable bend, can be chosen following the design goals. The applied
optics stability and magnet feasibility requirements constrain the cell
characteristics, starting from the drift lengths and the polarity of the
quadrupoles.

This analytical approach is used in order to define the appropriate initial
conditions for matching the lattice and for finding optimal regions of
operation for the best performance, subsequently verified numerically with
MADX. The CLIC DRs are chosen for numerically applying the concept of
including variable bends in the TME arc cells, and thereby demonstrate clearly
the reduction of the emittance as compared to the previous design, with
uniform dipoles. The emittance is so much reduced and lower with respect to
the target one, allowing the removal of a few TME cells. Moreover, by using
Nb3Sn high-field wigglers the final emittances including IBS are significantly
reduced, as well, resulting to the elimination of a few LSS FODO cells. As the
chromaticities are kept low, and the TME cell horizontal phase advance is
close to $\pi$, the new proposed design achieves excellent DA that can be
improved after a further working point optimization Ghasem  _et al._ (2016).
The alternative DR design achieves all target parameters (even for increased
bunch charge) although the circumference is reduced by around 13 %, as
compared to the original one.

##  VIII Acknowledgements

We are particularly indebted to two anonymous referees whose constructive
criticisms improved by far the clarity and quality of the paper. The research
leading to these results has received funding from the European Commission
under the FP7 Research Infrastructures project EuCARD-2, grant agreement
no.312453 and the European Union’s Horizon 2020 Research and Innovation
programme under Grant Agreement No 730871.

##  Appendix A Quadrupole focal lengths in the limit of small drift lengths

Following Eqs. (5), the full expressions of the quadrupole focal lengths,
$f_{1}$ and $f_{2}$, at the limits where the drifts $s_{1}$, $s_{2}$ and
$s_{3}$ are zeroed are:

  * •

$\lim_{s_{1}\to
0}f_{1}=\dfrac{s_{2}(\eta_{cd}+\tilde{\theta})}{\eta_{cd}+\tilde{\theta}-\eta_{ss1}+s_{2}\theta}~{}\mathrm{and}~{}\lim_{s_{1}\to
0}f_{2}=\dfrac{s_{2}\eta_{ss1}}{\eta_{cd}+\tilde{\theta}-\eta_{ss1}}$

  * •

$\lim_{s_{2}\to 0}f_{1}=0~{}~{}\mathrm{and}~{}~{}\lim_{s_{2}\to 0}f_{2}=0$

  * •

$\lim_{s_{3}\to
0}f_{1}=\dfrac{s_{2}g_{1}}{g_{1}+s_{2}\theta}~{}~{}\mathrm{and}~{}~{}\lim_{s_{3}\to
0}f_{2}=0$

where:

$\eta_{ss1}=\dfrac{\dfrac{-2(\eta_{cd}+\tilde{\theta})s_{3}}{s_{2}}}{1\pm\sqrt{1+\dfrac{4(\eta_{cd}+\tilde{\theta})s_{3}}{s_{2}^{2}}\dfrac{\beta_{cd}^{2}\theta-
g_{2}L/2}{\beta_{cd}^{2}\theta^{2}+g_{2}^{2}}}}$

and for $g_{1}$ and $g_{2}$ given by Eqs. (7).

##  Appendix B Integrals for the emittance of a variable bend

In order to calculate the emittance of a variable bend described by two
different bending radii functions along its length, Eq. (15) is used, with the
integrals $I_{1}$-$I_{8}$ being:

$I_{1}={\int\limits}_{0}^{L_{1}}\dfrac{\theta_{1}^{2}}{\left|\rho_{1}(s)\right|^{3}}ds\hskip
2.84544pt,\hskip
5.69046ptI_{2}={\int\limits}_{L_{1}}^{L_{1}+L_{2}}\dfrac{(\theta_{2}+\theta_{L_{1}})^{2}}{\left|\rho_{2}(s)\right|^{3}}ds\hskip
2.84544pt,\hskip
5.69046ptI_{3}={\int\limits}_{0}^{L_{1}}\dfrac{1}{\left|\rho_{1}(s)\right|^{3}}ds\hskip
2.84544pt,\hskip
5.69046ptI_{4}={\int\limits}_{L_{1}}^{L_{1}+L_{2}}\dfrac{1}{\left|\rho_{2}(s)\right|^{3}}ds\hskip
2.84544pt,\hskip
5.69046ptI_{5}={\int\limits}_{0}^{L_{1}}2\dfrac{-s\theta_{1}+\tilde{\theta}_{1}}{\left|\rho_{1}(s)\right|^{3}}ds$
(30)
$I_{6}={\int\limits}_{L_{1}}^{L_{1}+L_{2}}2\dfrac{-s\theta_{2}+\tilde{\theta}_{2}-L_{1}\theta_{L_{1}}+\tilde{\theta}_{L_{1}}}{\left|\rho_{2}(s)\right|^{3}}ds\hskip
2.84544pt,\hskip
5.69046ptI_{7}={\int\limits}_{0}^{L_{1}}\dfrac{(-s\theta_{1}+\tilde{\theta}_{1})^{2}}{\left|\rho_{1}(s)\right|^{3}}ds\hskip
2.84544pt,\hskip
5.69046ptI_{8}={\int\limits}_{L_{1}}^{L_{1}+L_{2}}\dfrac{(-s\theta_{2}+\tilde{\theta}_{2}-L_{1}\theta_{L_{1}}+\tilde{\theta}_{L_{1}})^{2}}{\left|\rho_{2}(s)\right|^{3}}ds\;\;,$

where $\theta_{L_{1}}=\theta(s=L_{1})$ and
$\tilde{\theta}_{L_{1}}=\tilde{\theta}(s=L_{1})$.

##  Appendix C Emittance reduction factor

As discussed in section III, the emittance reduction factors for the step and
the trapezium variable bend profiles depend only on $\tilde{\rho}$ and
$\lambda$ and their full expressions are given by:

$F_{TME_{step}}=2(\lambda+\tilde{\rho})^{3}(1+\tilde{\rho}^{2})\sqrt{\dfrac{1+\tilde{\rho}^{3}}{\lambda_{1}\tilde{\rho}^{4}+\lambda_{2}\tilde{\rho}^{5}+\lambda_{3}\tilde{\rho}^{7}+\lambda_{4}\tilde{\rho}^{8}+\lambda_{5}\tilde{\rho}^{9}+\lambda_{6}\tilde{\rho}^{10}+\lambda_{7}(1+9\tilde{\rho}^{3}+18\tilde{\rho}^{6})+\lambda_{8}(\lambda_{9}\tilde{\rho}^{11}+\lambda_{10}\tilde{\rho}^{12}+\tilde{\rho}^{13})}}~{},$

where $\lambda_{1}=4\lambda^{4}(5+18\lambda)$,
$\lambda_{2}=\lambda^{2}(9+45\lambda+64\lambda^{2})$,
$\lambda_{3}=12\lambda^{4}(5+21\lambda)$,
$\lambda_{4}=\lambda^{2}(31+210\lambda+399\lambda^{2})$,
$\lambda_{5}=\lambda(27+155\lambda+240\lambda^{2})$,
$\lambda_{6}=3(3+15\lambda+20\lambda^{2})$, $\lambda_{7}=4\lambda^{6}$,
$\lambda_{8}=(4+15\lambda+15\lambda^{2})$,
$\lambda_{9}=3\lambda_{7}/(4\lambda^{4})$ and
$\lambda_{10}=\lambda_{9}/\lambda$ .

$F_{TME_{trapezium}}=\dfrac{4\sqrt{2}r_{1}(r_{2}\lambda+\tilde{\rho}\ln\tilde{\rho})^{3}\sqrt{w_{2}/w_{3}}}{w_{1}}~{},$

where $r_{1}=\tilde{\rho}+1$, $r_{2}=\tilde{\rho}-1$ and:

$w_{1}=2r_{2}^{3}(2+3r_{1}\tilde{\rho})\lambda^{2}-6r_{2}\tilde{\rho}^{2}(r_{1}r_{2}-2\tilde{\rho}^{2}\ln\tilde{\rho})\lambda+3(r_{1}r_{2}-2\tilde{\rho}^{2}\ln\tilde{\rho})\tilde{\rho}^{3}+6\tilde{\rho}^{5}(\ln\tilde{\rho})^{2}~{},$
$w_{2}=r_{2}^{3}(2+r_{1}\tilde{\rho})w_{1}~{},$
$w_{3}=w_{3a}\lambda+w_{3b}\lambda^{2}+w_{3c}\lambda^{3}+w_{3d}\lambda^{4}+w_{3e}~{},$

for
$w_{3a}=90r_{2}\tilde{\rho}^{4}(r_{2}^{2}(-8+\tilde{\rho}(5+(-1+r_{2})\tilde{\rho}))+4(-1+r_{2})\tilde{\rho}\ln\tilde{\rho}(-2r_{2}+\tilde{\rho}\ln\tilde{\rho})),\\\
w_{3b}=15r_{2}^{2}\tilde{\rho}^{3}(r_{2}^{2}(20+3(2+r_{1})r_{2}\tilde{\rho})+4\tilde{\rho}^{2}\ln\tilde{\rho}(-2r_{2}+3(-1+r_{2})\ln\tilde{\rho})),w_{3c}=-120r_{2}^{4}\tilde{\rho}^{2}(r_{1}r_{2}-2\tilde{\rho}^{2}\ln\tilde{\rho}),\\\
w_{3d}=16r_{2}^{6}(1+3r_{1}\tilde{\rho})~{}\mathrm{and}~{}w_{3e}=45\tilde{\rho}^{5}(r_{2}^{2}(-44+\tilde{\rho}(-3+(-5+r_{2})\tilde{\rho}))+4\ln\tilde{\rho}(4r_{1}r_{2}+6r_{2}\tilde{\rho}+(-1+r_{2})\tilde{\rho}^{2}\ln\tilde{\rho}))$
.

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