# Equilibrium of an Arbitrary Bunch Train in the
Presence of Multiple Resonator Wake Fields

Robert Warnock<EMAIL_ADDRESS>SLAC National Accelerator
Laboratory, Stanford University, Menlo Park, CA 94025, USA  Department of
Mathematics and Statistics, University of New Mexico, Albuquerque, NM 87131,
USA

###### Abstract

A higher harmonic cavity (HHC), used to cause bunch lengthening for an
increase in the Touschek lifetime, is a feature of several fourth generation
synchrotron light sources. The desired bunch lengthening is complicated by the
presence of required gaps in the bunch train. In a recent paper the author and
Venturini studied the effect of various fill patterns by calculating the
charge densities in the equilibrium state, through coupled Haïssinski
equations. We assumed that the only collective force was from the beam loading
(wake field) of the harmonic cavity in its lowest mode. The present paper
improves the notation and organization of the equations so as to allow an easy
inclusion of multiple resonator wake fields. This allows one to study the
effects of beam loading of the main accelerating cavity, higher order modes of
the cavities, and short range geometric wakes represented by low-$Q$
resonators. As an example these effects are explored for ALS-U. The
compensation of the induced voltage in the main cavity, achieved in practice
by a feedback system, is modeled by adjustment of the generator voltage
through a new iterative scheme. Except in the case of a complete fill, the
compensated main cavity beam loading has a substantial effect on the bunch
profiles and the Touschek lifetimes. A $Q=6$ resonator, approximating the
effect of a realistic short range wake, is also consequential for the bunch
forms.

##  I Introduction

This is a sequel to Ref.prabI , in which we explored the action of a higher
harmonic cavity (HHC), a standard component of 4th generation synchrotron
light sources, employed to lengthen the bunch and reduce the effect of
Touschek scattering. In that work we introduced an effective scheme to compute
the equilibrium state of charge densities in an arbitrary bunch train. The
train is allowed to have arbitrary gaps and bunch charges. We chose the
simplest possible physical model, in which the only induced voltage (wake
field) is due to the lowest mode of the HHC. We write $V_{r3}$ for this
voltage, the notation designating “resonator, 3rd harmonic”. We recognized,
however, that excitation of the main accelerating cavity (MC) by the bunch
train produces an induced voltage $V_{r1}$ of comparable magnitude, the effect
described as beam loading. Our excuse for omitting $V_{r1}$ was that in
practice it is largely cancelled by adjusting the rf generator voltage $V_{g}$
through a feedback system. The sum of $V_{r1}$ and $V_{g}$ should closely
approximate $V_{rf}$, the desired accelerating voltage.

In real machines there are always gaps in the bunch train, and that leads to
varying bunch profiles and centroid displacements along the train. At first
sight this would suggest that $V_{r1}$ would be different for different
bunches, so that compensation could only be partial, perhaps only manifest in
some average sense. On the contrary, we shall calculate an equilibrium state
in which the compensation is essentially perfect for all bunches. This happens
by an adjustment of the charge densities of all bunches, to new forms that
sometimes differ substantially from those without the MC. The adjustment is
achieved automatically through a new algorithm presented here. This iterative
procedure minimizes a mean square deviation of $V_{r1}+V_{g}$ from $V_{rf}$,
summed over all bunches, as a function of two generator parameters, which are
equivalent to amplitude and phase.

It is not clear that this is a faithful model of the feedback mechanism, which
could conceivably amount to a weaker constraint on the bunch profiles.
Nevertheless, this study clarifies the mathematical structure of the problem,
and appears to be a worthwhile preliminary to a full time-dependent model of
the system including a realistic description of feedback.

Beside the main cavity, we also assess the role of the short range wake field
from geometric aberrations in the vacuum chamber, and higher order modes in
the HHC. These effects are added with the help of improvements in notation and
organization of the equations.

Extensive numerical results are reported with parameters for ALS-U. For
consistency with the previous work the parameters chosen are partially out of
date as the machine design stands at present, but that will not greatly affect
our pricipal conclusions. Although the qualitative picture of the model with
HHC alone is still in place, there are large quantitative changes. Even then,
we underestimate the full effects, because we can only get convergence of our
iterative method when the current is a few percent less than the design
current.

In Section II we briefly recall our previous algorithm for solving the coupled
Haïssinski equations. Section III introduces the improved notation and
organization which allows an easy inclusion of multiple resonator wake fields.
Section IV enlarges the system of equations to provide a calculation of the
diagonal terms in the potential, thus overcoming a limitation of the previous
formulation. Section V describes the method for determining the generator
parameters so as to compensate the induced voltage in the main cavity. Section
VI, with several subsections, reports numerical results for the case of ALS-U
alsu1 ; alsu2 , always making comparisons to results with only the HHC in
place. Subsection VI.1 treats the case of a complete fill, illustrating the
compensation of the main cavity in the simplest instance. Subsection VI.2
considers a partial fill with distributed gaps, as proposed for the machine.
Subsection VI.3 is concerned with over-stretching by reduction of the HHC
detuning. Subsection VI.4 explores the effect of the short range wake field
with a realistic wake potential. Subsection VI.5 checks the effect of the
principal higher order mode of the HHC. Subsection VI.6 presents our closest
approach to a realistic model, including the harmonic cavity, the compensated
main cavity, and the short range wake, altogether. Subsection VI.7 examines
the effect of the main cavity when there is only a single gap in the bunch
train. Section VII reviews our conclusions and possibilities for further work.
Appendix A derives the expression for the diagonal terms in the potential, for
resonators of arbitrary $Q$.

##  II Summary of method to compute the equilibrium charge densities

In prabI  we derived a set of equations to determine the equilibrium charge
densities of $n_{b}$ bunches, which may be stated succinctly as follows:

$F(\hat{\rho},I)=0\ .$ (1)

Here $I$ is the average current and $\hat{\rho}$ is a vector with $2n_{b}$
real components, consisting of the real and imaginary parts of
$\hat{\rho}_{i}(k_{r3})$, where $k_{r3}$ is the wave number of the lowest
resonant mode of the 3rd harmonic cavity. These quantities are defined in
terms of the beam frame charge densities $\rho_{i}(z)$, normalized to 1 on its
region of support $[-\Sigma,\Sigma]$, as

$\hat{\rho}_{i}(k)=\frac{1}{2\pi}\int_{-\Sigma}^{\Sigma}\exp\big{(}-ikz(1+i/2Q)\big{)}\rho_{i}(z)dz\
,$ (2)

where $Q$ is the quality factor of the cavity. The vector in (1) is arranged
as follows:

$\hat{\rho}=\big{[}~{}{\rm Re}\hat{\rho}_{1}(k_{r3}),\cdots,{\rm
Re}\hat{\rho}_{n_{b}}(k_{r3}),~{}{\rm Im}\hat{\rho}_{1}(k_{r3}),\cdots,{\rm
Im}\hat{\rho}_{n_{b}}(k_{r3})~{}\big{]}\ .$ (3)

Accordingly, $F$ in (1) is a real vector with $2n_{b}$ components, so that we
have $2n_{b}$ nonlinear algebraic equations in $2n_{b}$ unknowns, depending on
the parameter $I$.

For the high $Q$ of a typical HHC the quantity (2) is very close to the
Fourier transform, but we have persistently written all equations for general
$Q$ for later applications involving low-$Q$ resonators.

In (1) the diagonal terms of the induced voltage have been dropped, i.e. the
effects on a bunch of its own excitation of the cavity. This omission is
justified for the typical high $Q$ of an HHC. Our method to handle the
diagonal terms in the general case is introduced in Section IV.

A solution $\hat{\rho}$ of (1) determines the charge densities by the formula
of Eq.(50) in prabI ,

$\rho_{i}(z_{i})=\frac{1}{A_{i}}\exp\big{[}-\mu U_{i}(z_{i})\big{]}\ ,$ (4)

where $U_{i}$ is the potential felt by the $i$-th bunch, defined in Eq.(51) of
prabI . Here $\mu$ and $A_{i}$ are constants, and $z_{i}$ is the beam frame
longitudinal coordinate of the $i$-th bunch. The potential $U_{i}$ depends on
all components of $\hat{\rho}$, on the mean energy loss per turn $U_{0}$, and
on the parameters of the applied voltage $V_{rf}$ which we write as

$V_{rf}(z)=V_{1}\sin(k_{1}z+\phi_{1})=V_{1}\big{(}\cos\phi_{1}\sin(k_{1}z)+\sin\phi_{1}\cos(k_{1}z)\big{)}\
.$ (5)

We solve (1) by the matrix version of Newton’s iteration, defined in (67) of
prabI . We begin at small current $I$, taking all components of the first
guess for $\hat{\rho}$ to be the transform (2) of a Gaussian with the natural
bunch length. We then continue step-wise to the desired current, making a
linear extrapolation in current to provide a starting guess for the next
Newton iteration at incremented current. The extrapolation is accomplished by
solving for $\partial\hat{\rho}/\partial I$ from the $I$-derivative of (1):

$\frac{\partial F}{\partial\hat{\rho}}\frac{\partial\hat{\rho}}{\partial
I}+\frac{\partial F}{\partial I}=0\ .$ (6)

##  III Formalism for Multiple Resonators

The scheme allows the inclusion of any number of resonator wake fields, but to
do that conveniently requires some care in notation and organization of the
equations. With $n_{r}$ resonators there are $2n_{b}n_{r}=n_{u}$ unknowns,
which we assemble in one long vector $\tilde{\rho}$ :

$\displaystyle\tilde{\rho}=\big{[}~{}\tilde{\rho}(k)\ ,\
k=1,\cdots,n_{u}~{}\big{]}=$ $\displaystyle\big{[}~{}{\rm
Re}\hat{\rho}_{1}(k_{r1}),\cdots,~{}{\rm Re}\hat{\rho}_{n_{b}}(k_{r1}),~{}{\rm
Im}\hat{\rho}_{1}(k_{r1})~{},\cdots,~{}{\rm
Im}\hat{\rho}_{1}(k_{r1})~{},\cdots,$ $\displaystyle~{}~{}{\rm
Re}\hat{\rho}_{1}(k_{r,n_{r}}),\cdots,~{}{\rm
Re}\hat{\rho}_{n_{b}}(k_{r,n_{r}})~{},~{}{\rm
Im}\hat{\rho}_{1}(k_{r,n_{r}}),\cdots,~{}{\rm
Im}\hat{\rho}_{n_{b}}(k_{r,n_{r}})~{}\big{]}\ .$ (7)

Here $k_{r,n}$ is the resonant wave number of the $n$-th resonator, and the
subscript of $\hat{\rho}$ denotes as usual the bunch number.

To identify the bunch number and the resonator number for the $k$-th component
of the vector, we define two index maps: $\iota(k)$ which gives the bunch
number and $r(k)$ which gives the resonator number. Namely,

$\displaystyle\iota(k)=\left\\{\begin{array}[]{l}\mod(k,n_{b})~{}~{}{\rm
if}\mod(k,n_{b})\neq 0\\\ ~{}~{}~{}n_{b}~{}~{}{\rm if}\mod(k,n_{b})=0\\\
\end{array}\right\\}$ (10) $\displaystyle
r(k)=\bigg{\lceil}\frac{k}{2n_{b}}\bigg{\rceil}\ .$ (11)

Here $\lceil x\rceil$, the ceiling of $x$, is the least integer greater than
or equal to $x$. We also need two projection operators: $P_{re}(k)$ which is
equal to 1 if $k$ corresponds to a ${\rm Re}\hat{\rho}$ and is zero otherwise,
and $P_{im}(k)$ which is equal to 1 if $k$ corresponds to a ${\rm
Im}\hat{\rho}$ and is zero otherwise. These are expressed in terms of the
ceiling of $k/n_{b}$ as follows:

$\displaystyle P_{re}(k)=\frac{1}{2}\bigg{[}1-(-1)^{\lceil
k/n_{b}\rceil}\bigg{]}\ ,$ $\displaystyle
P_{im}(k)=\frac{1}{2}\bigg{[}1+(-1)^{\lceil k/n_{b}\rceil}\bigg{]}\ .$ (12)

The potential $U_{j}(z)$ for bunch $j$, generalizing Eq.(51) of prabI  to
allow $n_{r}$ resonators, is stated as

$\displaystyle
U_{j}(z)=\frac{eV_{1}}{k_{1}}\big{[}x_{1}\cos(k_{1}z)-x_{2}\sin(k_{1}z)-x_{1}\big{]}+U_{0}z$
(13)
$\displaystyle+\sum_{n=1}^{n_{r}}U^{d}_{jn}(z)+\sum_{k=1}^{n_{u}}M(z)_{j,k}~{}\tilde{\rho}_{k}\
,\quad j=1,\cdots,n_{b}\ ,\quad-\Sigma~{}\leq~{}z~{}\leq~{}\Sigma\ .$ (14)

The first term in (13) is $e$ times the integral of the applied voltage, now
called the generator voltage and written as

$V_{g}(z)=V_{1}\big{[}x_{1}\sin(k_{1}z)+x_{2}\cos(k_{1}z)\big{]}\ .$ (15)

At $x_{1}=\cos\phi_{1},\ x_{2}=\sin\phi_{1}$ this reduces to the desired
$V_{rf}$ of (5). In an amplitude-phase representation we have

$V_{g}(z)=\tilde{V}_{1}\sin(k_{1}z+\tilde{\phi}_{1})\
,\quad\tilde{V}_{1}=(x_{1}^{2}+x_{2}^{2})^{1/2}V_{1}\
,\quad\tilde{\phi}_{1}=\tan^{-1}(x_{2}/x_{1})\ .$ (16)

The first term in (14) represents the diagonal contributions, the effect on
bunch $j$ of its own excitation of the resonators, as opposed to excitation by
the other bunches which is described by the second term. By writing the latter
as a simple matrix-vector product we greatly simplify the calculation of the
Jacobian of the system, making it formally the same for any number of
resonators.

Referring to Eqs.(28), (51), (55), (56), (57), (58) of prabI , we can write
down the matrix elements $M(z)_{i,k}$ in the second term of (14). For this we
introduce a notation appropriate for labeling by the index $k$ of (7).
Functions of $k$, defined via the index maps, are labeled with a tilde:

$\displaystyle\tilde{k}_{r,k}=k_{r,r(k)}\ ,$
$\displaystyle\tilde{\xi}_{k}=\xi_{\iota(k)}\
,\quad\tilde{A}_{k}=A_{\iota(k)}$ $\displaystyle\tilde{R}_{sk}=R_{s,r(k)}\
,\quad\tilde{Q}_{k}=Q_{r(k)}\ ,$
$\displaystyle\tilde{\eta}_{k}=\eta_{r(k)},\quad\tilde{\psi}_{k}=\psi_{r(k)}\
,$
$\displaystyle\tilde{\phi}_{j,k}=\tilde{k}_{r,k}\big{[}(m_{\iota(k)}-m_{j})\lambda_{1}+\theta_{j-1,\iota(k)}C\big{]}\
,$ $\displaystyle\sigma_{j,k}(z)=\mathcal{S}\big{(}\tilde{k}_{r,k}z,\
\tilde{Q}_{k},\ \tilde{\phi}_{j,k}+\tilde{\psi}_{k}\big{)}\ ,$
$\displaystyle\gamma_{j,k}(z)=\mathcal{C}\big{(}\tilde{k}_{r,k}z,\
\tilde{Q}_{k},\ \tilde{\phi}_{j,k}+\tilde{\psi}_{k}\big{)}\ ,$ (17)

where

$\displaystyle\mathcal{S}(k_{r}z,Q,\phi)=\frac{1}{1+(1/2Q)^{2}}\bigg{[}\exp(-k_{r}z/2Q)\bigg{(}\sin(k_{r}z+\phi)-\frac{1}{2Q}\cos(k_{r}z+\phi)\bigg{)}\bigg{]}_{0}^{z}\
,$
$\displaystyle\mathcal{C}(k_{r}z,Q,\phi)=\frac{1}{1+(1/2Q)^{2}}\bigg{[}\exp(-k_{r}z/2Q)\bigg{(}\cos(k_{r}z+\phi)+\frac{1}{2Q}\sin(k_{r}z+\phi)\bigg{)}\bigg{]}_{0}^{z}\
.$ (18)

The result for the matrix from (51) and (57) of prabI  is seen to be (noting
that $\omega_{r}/k_{r}=c$)

$M(z)_{j,k}=2\pi
ce^{2}N\frac{\tilde{\eta}_{j}\tilde{R}_{sj}}{\tilde{Q}_{j}}(1-\delta_{j,\iota(k)})\tilde{\xi}_{k}\exp(-\tilde{\phi}_{j,k}/2\tilde{Q}_{k})\big{[}P_{re}(k)\sigma_{j,k}(z)+P_{im}(k)\gamma_{j,k}(z)\big{]}\
.$ (19)

In the present notation the system of coupled Haïssinski equations,
generalizing (66) of prabI , takes the form

$\displaystyle
F_{j}(\tilde{\rho})=\tilde{A}_{j}\tilde{\rho}_{j}-\frac{1}{2\pi}\int_{-\Sigma}^{\Sigma}\big{[}P_{re}(k)\cos(\tilde{k}_{r,j}\zeta)-P_{im}(k)\sin(\tilde{k}_{r,j}\zeta)\big{]}$
$\displaystyle\hskip
56.9055pt\cdot\exp\big{[}\tilde{k}_{r,j}\zeta/2\tilde{Q}_{j}-\mu\
U_{\iota(j)}(\zeta)\big{]}d\zeta=0\ ,\quad j=1,\cdots,n_{u}\ .$ (20)

The normalization integral appearing in the first term is

$\tilde{A}_{j}=\int_{-\Sigma}^{\Sigma}\exp\big{[}-\mu\
U_{\iota(j)}(\zeta)\big{]}d\zeta\ .$ (21)

We require the Jacobian matrix $[\partial F_{j}/\partial\tilde{\rho}_{k}]$ for
the solution of (20) by Newton’s method, assuming that the diagonal terms are
fixed. This is found immediately from (14), (20), and (21) as

$\displaystyle\frac{\partial
F_{j}}{\partial\tilde{\rho}_{k}}=\tilde{A}_{j}\delta_{j,k}-\mu\int_{-\Sigma}^{\Sigma}\exp\big{[}-\mu
U_{\iota(j)}(\zeta)\big{]}M(\zeta)_{j,k}$
$\displaystyle\cdot\bigg{[}\tilde{\rho}_{j}-\frac{1}{2\pi}\big{[}P_{re}(k)\cos(\tilde{k}_{r,j}\zeta)-P_{im}(k)\sin(\tilde{k}_{r,j}\zeta)\big{]}\exp\big{[}\tilde{k}_{r,j}\zeta/2\tilde{Q}_{j}\big{]}\bigg{]}d\zeta\
.$ (22)

The compact expressions in (19), (20), and (22) are quite convenient for
coding, and lead to a short program to solve the Haïssinski equations with any
number of resonators. For $\zeta$ at $n_{p}$ mesh points $z_{i}$ used in the
integrals we have the array $M(i,j,k)=M(z_{i})_{j,k}$ of manageable dimension
$n_{p}\times n_{b}\times n_{u}$ which can be computed and stored at the top,
outside the Newton iteration.

For the work of the following section we also need the induced voltage from
the main cavity, which we designate as the first resonator in the list
$(n=1)$. For the $j$-th bunch this takes the form

$\displaystyle V_{r1j}(z)=-2\pi
ceN\frac{k_{r1}R_{s1}\eta_{1}}{Q_{1}}\bigg{[}\sum_{k=1}^{2n_{b}}(1-\delta_{j,\iota(k)})\tilde{\xi}_{k}\exp(-(\tilde{k}_{r,k}z+\tilde{\phi}_{j,k})/2\tilde{Q}_{k})$
$\displaystyle\bigg{(}P_{re}(k)\cos(\tilde{k}_{r,k}z+\tilde{\phi}_{j,k}+\tilde{\psi}_{k})-P_{im}(k)\sin(\tilde{k}_{r,k}z+\tilde{\phi}_{i,k}+\tilde{\psi}_{k})\bigg{)}\tilde{\rho}_{k}+v^{d}_{1j}(z)\bigg{]}\
.$ (23)

The diagonal term $v^{d}_{1j}$ can be evaluated in terms of integrals derived
in Appendix A.

##  IV The full system of equations with diagonal terms

Through (20) we have a system of $n_{u}$ algebraic equations for determination
of $\tilde{\rho}$, provided that the diagonal terms in $U_{i}$ are given. The
latter are functionals of the charge densities $\rho_{i}(z_{i})$, from which
it follows that (20) can be stated in vector notation as

$\tilde{\rho}=\mathcal{A}(\tilde{\rho},\rho,I)\ .$ (24)

On the other hand, the $\rho_{i}(z_{i})$ are determined in turn as solutions
of integral equations provided that $\tilde{\rho}$ is given. The integral
equations are like normal single-bunch Haïssinski equations, but with a
background potential determined by $\tilde{\rho}$, namely

$\rho_{i}(z_{i})=\frac{1}{A_{i}}\exp\bigg{[}-\mu
U_{i}(z_{i},~{}\rho_{i},~{}\tilde{\rho})\bigg{]}\ ,\quad i=1,\cdots,n_{b}\ .$
(25)

In vector notation

$\rho=\mathcal{B}(\rho,\tilde{\rho},I)\ ..$ (26)

The potential $U_{i}$ depends on the $\rho_{i}$ through its diagonal terms, in
the first sum in (14). Our procedure will be to interleave the solution of
(24) at fixed $\tilde{\rho}$, by the usual Newton method, with the solution of
(26) at fixed $\tilde{\rho}$ . If this algorithm converges we shall have
consistency between $\rho$ and $\tilde{\rho}$ and a solution of the full
system.

It turns out, most fortunately, that the solution of (26) is obtained by plain
iteration as would be applied to a contraction mapping,

$\rho^{(n+1)}=\mathcal{B}(\rho^{(n)},\tilde{\rho},I)\ .$ (27)

In our application this usually converges to adequate accuracy in just one
step, or three at most, and takes negligible time.

This scheme based on (24) and (26) is used in all calculations reported below.
It replaces the method used in prabI , which was to evaluate the diagonal
terms from the value of $\rho$ from the previous Newton iterate. That works
only for high-$Q$ resonators, so is not adequate for handling the short range
machine wake.

##  V Algorithm to adjust the generator parameters $(x_{1},x_{2})$

We wish to choose $(x_{1},x_{2})$ so as to minimize, in some sense, the
difference

$V_{rf}(z_{i})-V_{g}(z_{i},x_{1},x_{2})-V_{r1i}(z_{i},x_{1},x_{2})\ ,$ (28)

for all $i=1,\cdots,n_{b}$. A reasonable and convenient choice for an
objective function to minimize is the sum of the squared $L^{2}$ norms of the
quantities (28). With a normalizing factor to make it dimensionless and of
convenient magnitude that is

$\displaystyle f(x_{1},x_{2})=$ $\displaystyle\frac{1}{2\Sigma
V_{1}^{2}}\sum_{i=1}^{n_{b}}\int_{-\Sigma}^{\Sigma}\bigg{[}V_{1}(\cos\phi_{1}-x_{1})\sin(k_{1}z)+V_{1}(\sin\phi_{1}-x_{2})\cos(k_{1}z)-V_{r1i}(z,x_{1},x_{2})\bigg{]}^{2}dz\
.$ (29)

The region of integration $[-\Sigma,\Sigma]$ is the same as that used in the
definition of the potential $U_{i}$.

Note that the minimum of $f$ cannot be strictly zero, since $V_{r1}$ is
sinusoidal with wave number $k_{r1}$, whereas the other terms are sinusoidal
with a slightly different wave number $k_{1}$.

Let us adopt the vector notation $x=(x_{1},x_{2})$ with norm
$|x|=|x_{1}|+|x_{2}|$. The equations to solve now depend on $x$, having the
form

$F(\tilde{\rho},I,x)=0\ .$ (30)

To avoid notational clutter we suppress reference to the diagonal terms,
leaving it understood that a solution of (30) for $\tilde{\rho}$ actually
involves the scheme of the previous session. As usual we solve for
$\tilde{\rho}$, for an increasing sequence of $I$-values. The scheme will be
to minimize $f(x)$ at each $I$, thus providing a new $x=\arg\min f$ to be used
at the next value of $I$. As will now be explained, the minimization will also
be done iteratively, so that we have an $x$-iteration embedded in the
$\tilde{\rho}$-iteration.

We wish to zero $\nabla_{x}F$, which is to find $x$ to solve the equations

$\displaystyle\sum_{i=1}^{n_{b}}\int_{-\Sigma}^{\Sigma}\bigg{[}V_{1}(\cos\phi_{1}-x_{1})\sin(k_{1}z)+V_{1}(\sin\phi_{1}-x_{2})\cos(k_{1}z)-V_{r1i}(z,x)\bigg{]}$
$\displaystyle\times\left[\begin{array}[]{c}V_{1}\sin(k_{1}z)+\partial_{x_{1}}V_{r1i}(z,x)\\\
V_{1}\cos(k_{1}z)+\partial_{x_{2}}V_{r1i}(z,x)\end{array}\right]dz=\left[\begin{array}[]{c}0\\\
0\end{array}\right]\ .$ (35) (36)

To solve (36) a first thought might be to apply Newton’s method, starting at
some low current and choosing the zero current solution
$(\cos\phi_{1},\sin\phi_{1})$ as the first guess. This would be awkward,
however, since it would involve the second derivatives of $V_{r1i}$ with
respect to $(x_{1},x_{2})$. The first derivatives must already be done by an
expensive numerical differentiation, and the second numerical derivative would
be error prone and even more expensive. Instead, let us assume that we have a
first guess $(x_{10},x_{20})$ and suppose that in a small neighborhood of that
point the first derivatives of $V_{r1i}$ can be regarded as constant. Then
second derivatives are zero and the Taylor expansion of $V_{r1i}$ gives two
linear equations to solve for $(x_{1},x_{2})$, namely

$\left[\begin{array}[]{cc}a_{11}&a_{12}\\\
a_{21}&a_{22}\end{array}\right]\left[\begin{array}[]{c}x_{1}\\\
x_{2}\end{array}\right]=\left[\begin{array}[]{c}b_{1}\\\
b_{2}\end{array}\right]\ ,$ (37)

where

$\displaystyle a_{11}=\sum_{i}\int\alpha_{1i}(z,x_{0})^{2}dz\ ,\quad
a_{22}=\sum_{i}\int\alpha_{2i}(z,x_{0})^{2}dz\ ,$ $\displaystyle
a_{12}=a_{21}=\sum_{i}\int\alpha_{1i}(z,x_{0})\alpha_{2i}(z,x_{0})dz\ ,$
$\displaystyle b_{1}=\sum_{i}\int\alpha_{1i}(z,x_{0})\beta_{i}(z,x_{0})dz\
,\quad b_{2}=\sum_{i}\int\alpha_{2i}(z,x_{0})\beta_{i}(z,x_{0})dz\ ,$ (38)

with

$\displaystyle\alpha_{1i}(z,x_{0})=V_{1}\sin(k_{1}z)+\partial_{x_{1}}V_{r1i}(z,x_{0})\
,$
$\displaystyle\alpha_{2i}(z,x_{0})=V_{1}\cos(k_{1}z)+\partial_{x_{2}}V_{r1i}(z,x_{0})\
,$
$\displaystyle\beta_{i}(z,x_{0})=-V_{r1i}(z,x_{0})+\nabla_{x}V_{r1i}(z,x_{0})\cdot
x_{0}+V_{1}\sin(k_{1}z+\phi_{1})$ (39) (40)

By (37) we have an update $x_{0}\rightarrow x$ which establishes the pattern
of the general iterate $x^{(k)}\rightarrow x^{(k+1)}$. This will be carried to
convergence in the sense $|x^{(k+1)}-x^{(k)}|<\epsilon_{x}$, with a suitable
$\epsilon_{x}$ to be determined by experiment. Each iterate requires a value
for $V_{r1i}$ and for $\nabla_{x}V_{r1i}$, which we compute numerically by a
divided difference,

$\frac{\partial V_{r1i}}{\partial
x_{1}}(z,x)\approx\frac{V_{r1i}(z,x_{1}+\Delta
x,x_{2})-V_{r1i}(z,x_{1},x_{2})}{\Delta x}\ .$ (41)

Thus one $x$-iteration requires three $\tilde{\rho}$-iterations to provide the
necessary values of $V_{r1i}$ (which are constructed from $\tilde{\rho}$). The
first $\tilde{\rho}$ iteration to find $V_{r1i}(z,x_{1},x_{2})$ produces a
$\tilde{\rho}$ which is a very good guess to start the remaining two
iterations to make the derivatives, which then converge quickly.

The choice of $\Delta x$ in (41) requires a compromise between accuracy and
avoiding round-off error. We found that $\Delta x=10^{-4}$ was widely
satisfactory, whereas success with smaller values depended on the
circumstances.

##  VI Numerical results with and without the main cavity

As in prabI  we illustrate with parameters for ALS-U alsu1 ; alsu2 , the
forthcoming Advanced Light Source Upgrade. Although the machine design is not
yet final, one provisional set of parameters for our main cavity (actually the
effect of two cavities together) is as follows:

$R_{s}=0.8259~{}M\Omega\ ,\quad Q=3486\ ,\quad\delta
f=f_{r1}-f_{1}=-82.54~{}kHz$ (42)

Here the shunt impedance $R_{s}$ and quality factor $Q$ are loaded values, the
unloaded values divided by $1+\beta$, with coupling parameter $\beta=7.233$.
We take these parameters for the main cavity, otherwise keeping the same
parameters as in prabI , Table I. Thus we take $U_{0}=$217 keV, even though a
value of 330 keV may be contemplated for the set (42).

###  VI.1 Complete Fill

We first take the case of a complete fill, thus $n_{b}=h=328$. The average
current is to be 500 mA, which we reach in 8 steps starting from 200 mA. The
CPU time is 15 minutes, rather than 20 seconds for the calculation without the
main cavity. The increase is mostly due to a much slower convergence of the
$\tilde{\rho}$-iteration, the $x$-iteration being a minor factor in CPU time.
To save time we gave $\epsilon_{x}$ the rather large value of 0.05, but then
made a refinement to $\epsilon_{x}=10^{-6}$ at the final current, in an extra
2 minute. The steepness of the objective function $f(x1,x2)$ of (29) is
extraordinary, having values around $10^{4}$ in the sequence with
$\epsilon_{x}=0.05$ while falling to a value close to 1 after the refinement.
An interesting question is how this steepness would be reflected in a feedback
system.

The result for the charge density, shown in the blue curve of Fig.1, is quite
close to the result without the main cavity, shown in red. It should be
emphasized that there is no explicit constraint requiring all bunches to be
the same. We have computed 328 bunches separately, and have found that they
all come out to be the same. This constitutes a good check on the correctness
of the equations and the code.

Figure 1: Charge density for complete fill at 500 mA, with compensated main
cavity (blue) and without main cavity (red).

In Fig.2 we show the compensation mechanism. The sum of the generator voltage
$V_{g}$ and the induced voltage $V_{r1}$ from the main cavity is the orange
curve. The latter deviates from the desired effective voltage $V_{rf}$ by less
than 2%, as is seen Fig.3.

Figure 2: MC induced voltage $V_{r1}$, generator voltage $V_{g}$ and their
sum. Figure 3: Relative deviation of $V_{r1}+V_{g}$ from $V_{rf}$.

The phasor of the generator voltage moves closer to $\pi/2$ and its magnitude
$\sqrt{x_{1}^{2}+x_{2}^{2}}$ increases from 1 to 1.0245, in comparison to the
phasor of $V_{rf}$. The corresponding values of $(x_{1},x_{2})$ are

$(x_{10},x_{20})=(\cos\phi_{1},\sin\phi_{1})=(~{}-0.93231,~{}0.36167~{})\quad\rightarrow\quad(x_{1},x_{2})=(~{}-0.29098,~{}0.98018~{})\
.$ (43)

###  VI.2 Partial fill C2 with distributed gaps

Next we take a partial fill with distributed gaps, labeled as fill C2; see
Section XIII-C of prabI . There are 284 bunches in 11 trains, with 4 empty
buckets between trains. There are 9 trains of 26 and 2 of 25, with the latter
positioned at opposite sides of the ring. All bunches have the same charge. As
in the preceding example we start the calculation at low average current and
advance in steps trying to reach the desired 500 mA. The convergence of
iterations is at first similar to that of the preceding case, but begins to
falter around 430 mA average current, at which point the convergence of the
$\tilde{\rho}$-iteration becomes problematic. By taking smaller and smaller
steps in current we can reach 496 mA, but beyond that point the Jacobian
matrix of the system appears to approach a singularity, as is indicated by its
estimated condition number having a precipitous increase, from 700 at the last
good solution to 2900 at a slightly higher current. Nevertheless, the
$x$-iterations continue to converge as long as the $\tilde{\rho}$-iterations
do. In the following, graphs are plotted for the maximum achievable current,
stated in figure captions.

Now the plots of $V_{g}$ and $V_{r1}$ and their sum look exactly the same as
in Fig.2, for every bunch. The minimization of $f(x_{1},x_{2})$ has caused the
bunch forms to rearrange themselves so that the compensation is essentially
perfect for every bunch. The deviation of $V_{r1}+V_{g}$ from $V_{rf}$,
scarcely visible on the scale of Fig.2, varies from bunch to bunch, but is
still less than 3% for all bunches.

Fig.5 shows 9 bunch profiles in one train, to be compared with the
corresponding results without the main cavity in Fig. 5. The main cavity
causes considerably more bunch distortion along the train, and also a bigger
variation in the rms bunch lengths, as is seen in Fig.6. The plots show the
ratio of bunch length to the natural bunch length. The head of the train is on
the right, with the highest bunch number.

The corresponding results for the bunch centroids is seen in Figs.7. Again the
deviation from the case without the main cavity is quite substantial.

Figure 4: Charge densities in a train of 26,
surrounded by gaps of 4 buckets, fill C2,
MC beam loading included, $I_{\rm av}=496$ mA.

Figure 5: Charge densities in a train of 26,
surrounded by gaps of 4 buckets, fill C2,
MC beam loading omitted, $I_{\rm av}=496$ mA.

Figure 6: Bunch length increases in a train of 26, surrounded by gaps of 4
buckets, with main cavity beam loading (blue) and without (red). $I_{\rm
av}=496$ mA. The plot is the ratio of bunch length $\sigma$ to the natural
bunch length $\sigma_{0}$. Figure 7: Centroids $<z>$ in a train of 26,
surrounded by gaps of 4 buckets, with cavity beam loading (blue) and without
(red). $I_{\rm av}=496$ mA

The main point of practical interest is the increase in Touschek lifetime
achieved through the bunch stretching caused by the HHC. Again, the MC has a
sizeable effect in reducing the lifetime and in causing a larger variation
along a train. This is shown in Fig.8 which gives the ratio of the lifetime
$\tau$ to the lifetime $\tau_{0}$ without the MC.

Figure 8: Touschek lifetime increase along a train, with compensated MC (blue)
and without (red). $I_{\rm av}=496$ mA.

We next consider the same fill pattern with 11 trains, but with a taper in the
bunch charges putting more charge at the ends, according to a power law as
shown in Fig. 15 of prabI . This is an example of invoking guard bunches to
reduce the effect of gaps. As is seen in Figs.10 and 10, the guarded inner
bunches, which resemble that of the complete fill, are little affected by the
MC. The strong asymmetry between the front and back of the train is perhaps
surprising, but it should be noticed that Fig.10 already shows an appreciable
front-back asymmetry. The strong amplification of this asymmetry by the MC is
in line with its big effects seen generally.

Figure 9: Case of tapered bunch charges,
MC beam loading included, $I_{\rm av}=496$ mA.

Figure 10: Case of tapered bunch charges,
MC beam loading omitted, $I_{\rm av}=496$ mA.

###  VI.3 Decrease of HHC detuning for over-stretching

There is practical interest in the possibility of over-stretching for an
additional increase in the Touschek lifetime. This entails a decrease in the
detuning of the HHC, which produces a larger r.m.s. bunch length but a bunch
profile with a dip in the middle, thus a double peak. In our case a decrease
from $df=250.2$ kHz to $df=235$ kHz produces a double peak in the model
without the MC at full current, as is seen in Fig.4 of prabI . We would like
to know how this setup looks with the compensated main cavity in play. Not
surprisingly, the convergence of our iterative solution breaks down at a lower
current than in the case of the normal detuning; the stronger the bunch
distortions the poorer the convergence. With $df=235$ kHz and the MC we can
only reach 474.5 mA, which is not enough to see a double peak. Nevertheless it
is useful to compare the result at that current with the result in absence of
the MC, as displayed for 9 bunches in a train of 27 in Figs.12 and 12.

Figure 11: Fill C2 with HHC + MC,
detuning $df=235$ kHz, $I_{\rm av}=474.5$ mA.

Figure 12: Fill C2 with HHC only,
detuning $df=235$ kHz, $I_{\rm av}=474.5$ mA.

Even at a current significantly less that the 500 mA design current the
distortion due to the main cavity is quite large, which leads to the
conclusion that the main cavity must be included in a realistic simulation of
over-stretching.

###  VI.4 Effect of the short range wake field

The short range wake field from various unavoidable corrugations in the vacuum
chamber retains importance in the latest storage rings, in spite of the best
efforts to reduce it. Since it can cause substantial bunch distortion in the
absence of an HHC, we would like to know how much it affects the operation of
the HHC. A result for the longitudinal wake potential at ALS-U, from a
detailed computation by Dan Wang dwang , is shown in Fig.13.

Figure 13: Wake potential (pseudo - Green function) for the ALS-U storage
ring, computed with a 1 mm driving bunch.

The corresponding impedance,

$Z(f)=\frac{1}{c}\int_{-\infty}^{\infty}e^{-ikz}W(z)dz\ ,\quad f=kc/2\pi\ ,$
(44)

is plotted in Fig.14.

Figure 14: Longitudinal impedance $Z(f)$ for ALS-U.

In Ref.prabI  we suggested that a low-$Q$ resonator wake could be treated on
the same footing as the high-$Q$ resonators, and for that reason we wrote all
equations for a general value of $Q$. We recognized, however, that the
diagonal term in the potential would now be dominant, while being nearly
negligible in the high-$Q$ case. It could not be treated by the method used in
prabI , but is easily handled by the presently adopted method of Section IV.

For the equilibrium state, the impedance at $f>20$ GHz is irrelevant, even
though it could have a role out of equilibrium. This assertion follows from
the fact that the frequency spectrum of our calculated charge densities never
extends beyond 15 GHZ, no matter which wake fields are included. Consequently,
a reasonable step is to concentrate on the first big peak at 11.5 GHz. The
wake potential in our equations (defined in (19) of prabI ) is based on an
impedance as follows, which is of Lorentzian form with half-width $\Gamma/2$:

$Z(f)=iR_{s}\frac{\Gamma}{2}\bigg{[}\frac{1}{f-f_{r}+i\Gamma/2}+\frac{1}{f+f_{r}+i\Gamma/2}\bigg{]}=Z(-f)^{*}\
,\quad\Gamma/2=f_{r}/2Q\ .$ (45)

Figures 16 and 16 show a fit to (45) with parameters as follows:

$f_{r}=11.549~{}{\rm GHz}\ ,\quad R_{s}=5730~{}\Omega\ ,\quad Q=6\ .$ (46)

The fit is rough in the imaginary part, but probably good enough to estimate
the magnitude of the effect of the short range wake.

Figure 15: Fit of ${\rm Re}Z$ to Lorentzian and
LRC circuit formulas.

Figure 16: Fit of ${\rm Im}Z$ to Lorentzian and
LRC circuit formulas.

Discussions of low-$Q$ resonator models in the literature usually invoke the
impedance of an LRC circuit, $Z(f)=R/(1+iQ(f_{r}/f-f/f_{r}))$, often with $Q$
near 1. As is illustrated in Figures 16 and 16, in our case with $Q=6$ the LRC
model does not give a better fit than the simpler Lorentzian, except for
enforcing $Z(0)=0$. At the expense of some complication our equations could be
modified to accommodate the LRC form, but that appears to be unnecessary, at
least in the present example.

Henceforth, the impedance from (45) and (46) will be referred to as SR (short
range). Taking first a complete fill, and including just the HHC and SR, we
get the result of Fig.17.

Figure 17: Charge density for a complete fill, with HHC plus the first peak in
the short range impedance (blue), and with HHC alone (red). $I_{\rm av}=500$
mA.

Next we consider the partial fill C2 with distributed gaps as treated in the
previous section. Figures 19 and 19 show the results for HHC+SR and HHC alone.

Figure 18: Fill C2, HHC + SR, $I_{\rm av}=476.2$ mA.

Figure 19: Fill C2, HHC alone, $I_{\rm av}=476.2$ mA.

As expected, the effects of SR are more pronounced in the partial fill than in
the complete fill. Correspondingly, the maximum current achieved is 472.6 mA.
As in previous cases we expect a substantially larger effect at the design
current of 500 mA.

###  VI.5 Higher order mode (HOM) of the harmonic cavity

At the present stage of design the most prominent longitudinal HOM of the HHC
for ALS-U is a TM011 mode with the following parameters luo :

$R_{s}=3000~{}\Omega\ ,\quad Q=80\ ,\quad f_{r}=2.29~{}{\rm GHz}$ (47)

A calculation for fill C2 with the HHC and this HOM gave the result of Fig.20.
The effect of the HOM on the charge densities is less than 2%, in a small
shift at the top of the distributions.

At least for the equilibrium state in ALS-U, it appears that the HOM can be
neglected. The role of HOM’s in longitudinal coupled-bunch instabilities is
discussed in Ref.cullinan .

Figure 20: Two bunches in fill C2 with HHC and its higher order mode. $I_{\rm
av}=500$ mA.

###  VI.6 The full model: HHC+MC+SR.

We are now prepared to include the harmonic cavity, the compensated main
cavity, and the short range wake, altogether. The convergence of the Newton
sequence suffers even more than in the previous cases, and the continuation in
current reaches only $I_{\rm av}=471.9$ mA. Effects seen at this current must
severely underestimate what can be expected at 500 mA, because of the strong
variation near the design current that we have observed in every case.

For fill C2 we see the charge densities in Figures 22 and 22.

Figure 21: HHC+MC+SR, $I_{\rm av}=471.9$ mA.

Figure 22: HHC+MC, $I_{\rm av}=471.9$ mA.

###  VI.7 The case of a single gap, with main cavity beam loading

It is worthwhile to examine the effect of main cavity beam loading when there
is only a single gap in the fill pattern, even though this is not directly
relevant to the ALS-U design. With 284 bunches, a gap of 44 buckets, and
HHC+MC we get the result of Fig.24 for charge densities, to be compared with
the case of HHC alone in Fig.24. This result could be obtained with the full
current of $500$ mA. The graphs show 6 bunches at the head of the train
(right), middle of the train (middle), and end of the train (left).

Figure 23: HHC+MC, single gap, $I_{\rm av}=500$ mA.

Figure 24: HHC, single gap, $I_{\rm av}=500$ mA.

The bunch lengthening is smaller and the centroid displacement greater when
the MC is included. The comparison of bunch lengthenings is shown in Fig.25.

Figure 25: Single gap, bunch lengthening ratio, for HHC+MC (blue) and with HHC
alone (red).

##  VII Conclusions and outlook

Continuing the investigation of Ref.prabI  we have extended the physical model
to include the effect of the main accelerating cavity in its fundamental mode,
previously omitted. We introduced a new algorithm to adjust the parameters of
the rf generator voltage so as to compensate the voltage induced in the cavity
by the beam, thus putting the net accelerating voltage at a desired value.
When the cavity is excited by a bunch train with gaps this compensation
implies a modification of the bunch profiles, which is produced automatically
in our scheme.

We illustrated the outcome for parameters of the forthcoming ALS-U storage
ring, revisiting examples treated in prabI  without the main cavity. The
results are similar in modo grosso, but there are significant quantitative
differences, especially in cases of overstretching of bunches. Generally
speaking there is more bunch distortion and less symmetrical patterns in the
bunch trains, and the rms bunch lengthening is a bit smaller and much more
variable along the train. Correspondingly, the Touschek lifetime increase is
smaller and more variable over a train.

We have not tried to model the feedback system that compensates the beam
loading in practice. Our aim was only to show the theoretical existence of an
equilibrium state with precise compensation in place.

It was disappointing, and somewhat surprising, to find that the Newton
iteration to solve the coupled Haïsinski equations encounters convergence
difficulties at large current (near the design current) when either the main
cavity wake or the short range wake is added to the HHC wake.

A colleague suggested that the failure of convergence might hint at an
instability. One should be cautious about such an idea. The issue here is just
the existence of an equilibrium. An equilibrium may or may not be stable under
time evolution, so stability is a different issue.

Our failure to find an equilibrium in some cases of high current may be due to
a failure of technique, not necessarily an indication that no equilibrium
exists. At high current we are trying to achieve convergence of the Newton
iteration close to a singularity of the Jacobian, but not squarely on the
singularity. In this case it is crucial to have a starting guess sufficiently
close to a solution, but in practice the required degree of closeness is
unknown. We made some efforts to improve the guess by a seemingly careful
continuation in current from the last good solution, but there was no clear
success.

A likely remedy for the convergence failure is to return to the conventional
formulation of the Haïssinski equations as integral equations for the charge
densities, in place of the present formulation as algebraic equations for
Fourier amplitudes. For a single Haïssinski integral equation discretized on a
mesh in $z$-space, the Newton iterative solution is ultra-robust, converging
at currents far beyond realistic values bobkarl . It seems likely that similar
good behavior will hold for the coupled integral equations. The size of the
discretized system does not grow with the number of resonator wakes, in
contrast to the present system, and the full $z$-space description of the
short range wake could be invoked in place of the low-$Q$ resonator model.

To make this $z$-space formulation feasible on modest computer resources we
can assume that all bunch sub-trains are identical, and all separated by
identical gaps. For the ALS-U this would mean artificially increasing the
harmonic number from 328 to 330, and having 11 trains of 26 separated by gaps
of 4 buckets. Then we have 26 independent charge densities, which can
adequately be described by 100 mesh points each. Thus the Jacobian of the
Newton iteration is $2600\times 2600$, a modest size that will yield a very
quick computation.

Moreover, this identical train model would make it feasible to do a time-
domain solution of the coupled Vlasov-Fokker-Planck equations by the method of
local characteristics (discretizing the Perron-Frobenius operator) senigallia
. This could answer the urgent question of stability of the equilibria, and
provide a window to the dynamics out of equilibrium. Incidentally,
determination of the instability threshold by the linearized Vlasov system
could also be attempted.

It was gratifying to find that the sub-iteration to enforce the main cavity
compensation converged very quickly whenever the main iteration converged. It
can be employed in the same way in the proposed $z$-space system. Also, the
formalism for multiple resonator wakes will still be advantageous in the
$z$-space scheme.

##  VIII Acknowledgments

I thank Teresia Olsson for a helpful correspondence, Dan Wang for her wake
potential, and Tianhuan Luo for information on the HHC design. Marco Venturini
posed the main cavity compensation problem in general terms. Karl Bane
encouraged the study of the short range wake. This work was supported in part
by the U. S. Department of Energy, Contract Nos. DE-AC03-76SF00515. My work is
aided by an affiliation with Lawrence Berkeley National Laboratory as Guest
Senior Scientist.

##  Appendix A Diagonal terms in the potential.

Here we find the formula for a generic term in the first sum of (14). For this
we revert to the notation used in the case of a single resonator.

The term in question is the last term of (51) in prabI , defined through (60)
of that paper, as follows:

$\displaystyle
U_{i}^{d}(z_{i})=\frac{e^{2}N\omega_{r}R_{s}\eta\xi_{i}}{Q}\bigg{[}\int_{0}^{z_{i}}d\zeta\int_{-\Sigma}^{\zeta}\exp(-k_{r}(\zeta-u)/2Q)\cos(k_{r}(\zeta-u)+\psi)\rho_{i}(u)du$
$\displaystyle+\int_{0}^{z_{i}}d\zeta\int_{\zeta}^{\Sigma}\exp(-k_{r}(\zeta-u+C)/2Q)\cos(k_{r}(\zeta-u+C)+\psi)\rho_{i}(u)du\bigg{]}\
.$ (48)

The repeated integrals can be replaced by single integrals through integration
by parts. First apply the double angle formula to the cosine, so as to bring
out factors of $\cos(k_{r}u)$ and $\sin(k_{r}u)$. The $u$-integrals involving
those factors are functions of $\zeta$, which are to be differentiated in the
partial integration with respect to $\zeta$. The corresponding integration
with respect to $\zeta$ is done with the help of (55) and (56) (as indefinite
integrals) in prabI . The result is

$\displaystyle
U^{d}_{i}(z_{i})=\frac{ce^{2}NR_{s}\eta\xi_{i}}{Q(1+(1/2Q)^{2})}\big{[}~{}I_{1}+I_{2}~{}\big{]}\
,$ $\displaystyle
I_{1}=\int_{-\Sigma}^{z_{i}}\exp(k_{r}u/2Q)\bigg{[}a(z_{i})\cos(k_{r}u)+b(z_{i})\sin(k_{r}u)\bigg{]}\rho_{i}(u)du$
$\displaystyle\hskip
22.76228pt-\big{(}\sin\psi-\frac{1}{2Q}\cos\psi\big{)}\int_{-\Sigma}^{z_{i}}\rho_{i}(u)du\
,$ $\displaystyle
I_{2}=\int^{\Sigma}_{z_{i}}\exp(k_{r}u/2Q)\bigg{[}a(z_{i}+C)\cos(k_{r}u)+b(z_{i}+C)\sin(k_{r}u)\bigg{]}\rho_{i}(u)du$
$\displaystyle\hskip
22.76228pt+\exp(-k_{r}C/2Q)\big{(}\sin(k_{r}C+\psi)-\frac{1}{2Q}\cos(k_{r}C+\psi)\big{)}\int_{-\Sigma}^{z_{i}}\rho_{i}(u)du\
,$ $\displaystyle
a(z)=\exp\big{(}-k_{r}z/2Q\big{)}\big{(}\sin(k_{r}z+\psi)-\frac{1}{2Q}\cos(k_{r}z+\psi)\big{)}\
,$ $\displaystyle
b(z)=-\exp\big{(}-k_{r}z/2Q\big{)}\big{(}\cos(k_{r}z+\psi)+\frac{1}{2Q}\sin(k_{r}z+\psi)\big{)}\
.$ (49)

Here we have dropped and added terms independent of $z_{i}$, which only affect
the normalization (21), and have used the double angle formula in reverse to
consolidate some terms. Writing
$\int_{z_{i}}^{\Sigma}=\int_{-\Sigma}^{\Sigma}-\int_{-\Sigma}^{z_{i}}$, we see
that there are three different integrals to evaluate,

$\int_{-\Sigma}^{z_{i}}\big{[}~{}1,~{}\exp(k_{r}u/2Q)\cos(k_{r}u),~{}\exp(k_{r}u/2Q)\sin(k_{r}u)~{}\big{]}\rho_{i}(u)du\
,$ (50)

which can be built up stepwise on a mesh in $z_{i}$. Thus we can compute and
store the diagonal terms on the mesh in negligible time. Note that $I_{2}$ is
totally negligible for the small $Q$ that we encounter in representing the
geometric wake, owing to the tiny prefactor $\exp(-k_{r}C/2Q)$.

Summing (49) over the $n_{r}$ choices of the resonator parameters
$k_{r},R_{s},Q,\eta,\psi$ we obtain the first term of (14).

## References

  * (1) R. Warnock and M. Venturini, Equilibrium of an arbitrary bunch train in presence of a passive harmonic cavity: Solution through coupled Haïssinski equations, Phys. Rev. Accel. Beams 23, 064403 (2020).
  * (2) C. Steier, A. Anders, J. Byrd, K. Chow, R. Duarte, J. Jung, T. Luo, H. Nishimura, T. Oliver, J. Osborn et al., “R+D progress towards a diffraction limited upgrade of the ALS”, Proc. IPAC2016, Busan, Kroea.
  * (3) C. Steier, A. Allézy, A. Anders, K. Baptiste, J. Byrd, K. Chow, G. Cutler, R. Donahue, R. Duarte, J.-Y. Jung et al., “Status of the conceptual design of ALS-U”, Proc. IPAC2017, Copenhagen, Denmark.
  * (4) Dan Wang, Lawrence Berkeley National Laboratory, private communication. This is from work in progress.
  * (5) Tianhuan Luo, Lawrence Berkeley National Laboratory, private communication.
  * (6) F. J. Cullinan, Å. Andersson, and P. F. Tavares, Harmonic-cavity stabilization of longitudinal coupled-bunch instabilitiess with a nonuniform fill, Phys. Rev. Accel. Beams 23, 074402 (2020).
  * (7) R. Warnock and K. Bane, Numerical Solution of the Haïssinski Equation for the Equilibrium State of a Stored Electron Beam, Phys. Rev. Accel. Beams 21, 124401 (2018).
  * (8) R. Warnock, Study of Bunch Instabilities by the Nonlinear Vlasov-Fokker-Planck Equation, Nuc. Instrum. Methods Phys. Res. A 561, 186 (2006).


# Silicon carbide soliton microcomb generation for narrow-grid optical
communications

Jingwei Li  1  Ruixuan Wang 1  Haipeng Zheng 2  Zhensheng Jia *2  and Qing
Li*1  1Department of Electrical and Computer Engineering, Carnegie Mellon
University, Pittsburgh, PA 15213, USA  2 CableLabs, 858 Coal Creek Circle,
Louisville, Colorado 80027, USA<EMAIL_ADDRESS><EMAIL_ADDRESS>
###### Abstract

A soliton microcomb can play a crucial role in narrow-grid optical
communications by replacing many independently operated lasers in wavelength-
division multiplexing systems. In this work, we designed and demonstrated
power-efficient soliton microcombs with 100-GHz free spectral range in an
integrated 4H-SiC platform for the first time. The combination of enabling
technologies, including efficient fiber coupling (3 dB insertion loss), high-
quality-factor microrings (intrinsic quality factors up to 5.7 million), and
the employment of the Raman effect for adiabatic accessing of the soliton
state, has enabled the demonstration of soliton pump power as low as 6 mW
while supporting comb powers above -20 dBm per line near the pump wavelength.

††journal: osajournal††articletype: article

##  1 Introduction

A soliton microcomb source consists of multiple comb lines that are equally
spaced and phase coherent with each other, which has revolutionized the
traditional frequency comb technologies by reducing the device’s size, weight,
and power (SWaP) while enabling system-level integration [1]. Soliton
microcombs have found a wide range of applications such as frequency
generation and synthesis [2, 3, 4, 5], imaging and sensing [6, 7, 8, 9], light
detection and ranging [10, 11], parallel optical communication and computation
[12, 13], and quantum information processing [14, 15]. Several integrated
photonic platforms, including silicon [16, 17], silica [18, 19], aluminum
nitride [20, 21], silicon nitride [22, 23], lithium niobate [24, 25, 26],
AlGaAs [27, 28], have shown promises for microcomb generation.

In this work, we exploit an emergent integrated photonics platform,
4H-silicon-carbide-on-insulator (4H-SiCOI), for producing power-efficient
soliton microcombs suitable for narrow-grid optical communications [29, 30,
31]. With tailored properties, such a microcomb source could play a crucial
role in network facilities, such as central offices or hubs, by replacing many
independently operated lasers in wavelength-division multiplexing (WDM)
systems, thereby significantly reducing the overall cost and power consumption
of optical hardware [32]. The 4H-SiCOI platform is selected mainly because of
its strong Kerr nonlinearity, which is estimated to be four times of that of
silicon nitride and thus allows a reduction of pump power by more than one
order of magnitude (assuming other factors are the same) [33]. Our device
optimization has resulted in the first demonstration of single-soliton SiC
microcomb with a free spectral range of 99.8 GHz, operable for pump powers as
low as 6 mW. In addition, associated enabling technologies, such as efficient
fiber coupling and the utilization of Raman-active resonances for
adiabatically accessing the soliton state, have also been developed.

##  2 Device design and fabrication

As illustrated in Fig. 1(a), the 4H-SiCOI integrated photonics platform
consists of a thin-layer of 4H-SiC on top of lower-index cladding materials
such as silicon dioxide. In this work, the SiC thickness is chosen to be 700
nm for the demonstration of microcombs tailored for narrow-grid optical
communications. Using optimized nanofabrication, we obtain low-loss photonic
devices, including waveguides and microresonators, with varied geometries
(details see Ref. [34]). Figure 1(b) shows one such example, i.e., a
169-$\mu$m-radius SiC microring with a ring waveguide width of 3 $\mu$m. Its
free spectral range (FSR) is expected to be near 100 GHz, a preferred choice
for the alignment of the produced comb lines with the dense WDM International
Telecommunication Union (ITU) grids. In Fig. 1(c), the simulated dispersion of
the fundamental transverse-magnetic (TM00) and transverse-electric (TE00)
modes is provided, where the integrated dispersion ($D_{\text{int}}$) is
defined as: $D_{\text{int}}\equiv\omega_{\mu}-\omega_{0}-D_{1}\mu$, with $\mu$
being the relative azimuthal order to the pump resonance (i.e., $\mu=0$ for
the pump mode), $\omega_{\mu}$ being the corresponding resonance frequency,
and $D_{1}/(2\pi)$ corresponding to the FSR. As can be seen, the TM00 mode
exhibits a relatively strong anomalous dispersion, with its dominant group
velocity dispersion (GVD) computed to be near 400
ps/($\text{nm}\cdot\text{km}$). In contrast, the GVD of the TE00 mode is still
anomalous but much weaker. Compared to the TE00 mode, the TM00 resonance is
likely to yield a higher average power per comb line, albeit at the cost of a
smaller comb bandwidth. This prediction is confirmed by numerical simulations
based on the Lugiato-Lefever equation (LLE) [35]. For example, with an
injection power of 5 mW to the TM00 resonance, the power of comb lines
adjacent to the pump is estimated to be larger than -20 dBm (10 $\mu$W, see
Fig. 1(d)), which is sufficient for further amplification and data
transmission in optical communications.

Figure 1: (a) Illustration of a 4H-silicon-carbide-on-insulator (4H-SiCOI)
waveguide (SiC thickness of 700 nm and a pedestal layer of 150 nm) and the
mode profile of the fundamental transverse-magnetic (TM00) mode. (b) Scanning
electron micrograph of a 169-$\mu$m-radius SiC microring (ring width of $3\
\mu$m) with an expected free spectral range (FSR) of 100 GHz. (c) Simulated
integrated dispersion ($D_{\text{int}}$) of the fundamental transverse-
electric (TE00) and TM00 mode families in the 1550 nm band. (d) Simulated
optical spectrum of the single-soliton microcomb corresponding to the TM00
mode of a 100-GHz-FSR microring with an intrinsic (coupling) $Q$ of 5 (5)
million. The Kerr nonlinear parameter and the on-chip power are assumed to be
$3.2\ \text{W}^{-1}\text{m}^{-1}$ and 5 mW, respectively.

##  3 Fiber coupling and linear testing

Figure 2: (a) Optical micrograph of a lensed fiber coupled to an inverse taper
implemented on the SiC chip. (b) Linear transmission scan of a
169-$\mu$m-radius SiC microring for the TM polarization, in which the TM00
mode is dominantly excited. The total insertion loss is estimated to be within
6-7 dB range. (c) Representative resonance of the TM00 mode with normalized
transmission in the 1550 nm band, showing a loaded and intrinsic quality
factors of $2.9$ million and $4.4$ million, respectively. (d) Dispersion
characterization of the TM00 mode families, displaying an approximate group
velocity dispersion of 400 $\text{ps}/(\text{nm}\cdot\text{km})$. (e)
Representative resonance of the TE00 mode with normalized transmission in the
1550 nm band, showing a loaded and intrinsic quality factor of $5.0$ million
and $5.7$ million, respectively. (f) Dispersion characterization of the TE00
mode families, displaying an approximate group velocity dispersion of 20
$\text{ps}/(\text{nm}\cdot\text{km}$).

To minimize the coupling loss between the fiber and the SiC chip, we taper
down the SiC waveguide width near the facets to increase the modal field
overlap between the waveguide and a lensed fiber (see Fig. 2(a)). Numerical
simulations point to a taper width of 250 nm to be best matched to a lensed
fiber with a mode field diameter of $2.5\ \mu$m. In practice, the insertion
loss is measured to be slightly higher than the theoretical value (2 dB, or
$\approx 60\%$ coupling efficiency), close to 3 dB ($\approx 50\%$ coupling
efficiency) per facet instead (see Fig. 2(b)). In addition, we employ a pulley
coupling scheme (see Fig. 1(b)) to selectively excite the desired resonant
modes, which only allows the phase-matched resonances to be efficiently
coupled. Take the transmission scan plotted in Fig. 2(b) as an example: it
consists of a dominant excitation of the TM00 mode families despite the
multimode nature of the 3-$\mu$m-wide ring waveguide. For a coupling gap of
350 nm, the pulley-coupled TM00 resonance is slightly under-coupled,
displaying an intrinsic quality factor ($Q$) in the $3.5$-$4.5$ million range
(see Fig. 2(c) for one instance). The coupling level becomes stronger as we
move toward longer wavelengths, which is evidenced by the increased extinction
ratio in the resonance scan (see Fig. 2(b)). On the other hand, the TE00 modes
are severely under-coupled, with the extracted intrinsic $Q$s in the $5$-$6$
million range (see Fig. 2(e) for one instance). Finally, we can characterize
the resonator’s dispersion by calibrating the recorded resonance frequencies
with the aid of a wavemeter. The results provided in Fig. 2(d) and Fig. 2(f),
corresponding to the respective dispersion of TM00 and TE00 mode families, are
consistent with simulation data in Fig. 1(c). We also note that while the TM00
resonances have shown minimum frequency deviation from the expected dispersion
(Fig. 2(d)), several avoided mode crossings are observed for the TE00
resonances (Fig. 2(f)). Such frequency shifts for the TE00 mode may negatively
impact the microcomb formation, which will be discussed later.

##  4 Nonlinear testing

When increasing the pump power to the mW level, optical parametric oscillation
(OPO) is observed for both TE00 and TM00 resonances due to their anomalous GVD
and high optical $Q$s. We can estimate the threshold power of OPO as [36]

$P_{\text{th, OPO}}\approx\frac{\pi
n_{g}^{2}V_{\text{eff}}}{4\lambda_{p}n_{2}}\cdot\frac{Q_{c,p}}{Q_{l,p}^{3}},$
(1)

where $n_{g}$ is the group index ($n_{g}\approx 2.8$); $V_{\text{eff}}$ is the
effective mode volume ($V_{\text{eff}}\approx 1290\ {\mu m}^{3}$ for
169-$\mu$m-radius SiC microrings); $\lambda_{p}$ denotes pump wavelength;
$n_{2}$ is the Kerr nonlinear index of 4H-SiC; and $Q_{c,p}$ and $Q_{l,p}$ are
the coupling $Q$ and loaded $Q$ of the pump resonance, respectively. This
relationship allows us to connect the Kerr nonlinear index of 4H-SiC to the
observed OPO threshold power. For example, the OPO threshold of the TM00
resonance near $1545.5$ nm is measured to be $2.7$ mW (see Fig. 3(a)), while
its coupling and loaded $Q$ values are found to be $8.3$ million and $2.55$
million, respectively. Applying Eq. 1, we infer a Kerr nonlinear index of
$n_{2}\approx 9.5\times 10^{-19}\text{m}^{2}/\text{W}$. To account for the
variations of the $Q$ factors among different azimuthal orders, we carry out
the same measurement on other TM00 and TE00 resonances, and plot the extracted
$n_{2}$ in Fig. 3(c). The statistically averaged Kerr nonlinear index index is
found to be $n_{2}^{\text{TM}}\approx(9.4\pm 0.4)\times
10^{-19}\text{m}^{2}/\text{W}$ for the TM polarization and
$n_{2}^{\text{TE}}\approx(8.8\pm 0.4)\times 10^{-19}\text{m}^{2}/\text{W}$.
Note that these numbers are consistent with the approach based on the four-
wave mixing experiment carried out in a separate work [33].

Figure 3: (a) Optical spectrum showing the optical parametric oscillation
(OPO) of the TM00 resonance when pumped near $1545.5$ nm, with its OPO
threshold power estimated to be near $2.7$ mW. (b) Optical spectrum showing
stimulated Raman scattering of the TM00 resonance when pumped near 1556 nm.
The Raman threshold power is near $1.7$ mW, with the Stokes signal generated
approximately $23.3$ THz away from the pump laser (corresponding to a Raman
shift of $777\ \text{cm}^{-1}$). (c) Estimated Kerr nonlinear refractive index
of 4H-SiC based on the measured OPO threshold power for various TM00 and TE00
resonances near 1550 nm.

For high-$Q$ SiC microresonators, however, there is a competing Raman effect
that exhibits a comparable threshold power as that of OPO [37]. In 4H-SiC, the
dominant Raman frequency shift is around 777 $\text{cm}^{-1}$ (or $23.3$ THz)
[31]. The threshold power corresponding to the stimulated Raman scattering
process is given by [38]

$P_{\text{th,
Raman}}\approx\frac{\pi^{2}n_{g}^{2}V_{\text{eff}}}{\lambda_{p}\lambda_{s}g_{R}}\cdot\frac{Q_{c,p}}{Q_{l,p}^{2}}\cdot\frac{1}{Q_{l,s}},$
(2)

where $\lambda_{s}$ and $Q_{l,s}$ denote the wavelength and loaded $Q$ of the
Stokes resonance, respectively; and $g_{R}$ is the Raman gain coefficient
which has a peak value of $g_{R}^{\text{max}}=(0.75\pm 0.15)\
\text{cm}/\text{GW}$ in the 1550 nm band and an approximate full width at half
maximum (FWHM) of ($120\pm 30$) GHz [38]. One can understand the competition
between the Raman and OPO effects in 4H-SiC by comparing their respective
threshold powers as

$\frac{P_{\text{th,Raman}}}{P_{\text{th,OPO}}}=\frac{4\pi
n_{2}}{g_{R}\lambda_{s}}\cdot\frac{Q_{l,p}}{Q_{l,s}}\approx
0.85\cdot\frac{g_{R}^{\text{max}}}{g_{R}}\cdot\frac{Q_{l,p}}{Q_{l,s}}.$ (3)

According to Eq. 3, we deduce that if the FSR of the microresonator is much
larger than the FWHM bandwidth of the Raman gain ($\approx 120$GHz), $g_{R}$,
the actual Raman gain of the Stokes resonance, is likely to be much smaller
than $g_{R}^{\text{max}}$. This is because unless there is accidental
frequency matching, the Stokes resonance is typically misaligned to the center
of the Raman gain spectrum. As a result, the Kerr effect has a high
probability to dominate over the Raman process. On the other hand, when the
FSR of the microresonator is reduced below the FWHM of the Raman gain, which
is the case here, $g_{R}$ becomes comparable to $g_{R}^{\text{max}}$ (for
100-GHz FSR, the worst $g_{R}$ is $60\%$ of $g_{R}^{\text{max}}$). The
dominant nonlinear effect, whether it is Raman or Kerr, hence depends on the
exact frequency mismatching condition and the ratio of loaded $Q$s between the
pump and Stokes resonances [39]. In fact, for the 169-$\mu$m-radius SiC
microring, the TM00 mode families are observed to exhibit two distinct
behaviors. For resonances below 1550 nm, the OPO process appears first, such
as the one shown in Fig. 3(a). In contrast, for certain resonances above 1550
nm, such as the 1556 nm resonance shown in Fig. 3(b), the Stokes Raman
dominates over the Kerr effect with a lower power threshold. For this type of
resonance, the Kerr effect is still strongly suppressed even after increasing
the pump power above the OPO threshold (i.e., no microcomb generation) [37].

##  5 Adiabatic accessing of soliton states

Figure 4: (a) Experimental schematic for adiabatically accessing the single-
soliton state with the aid of an auxiliary cooling laser. (b) Pump scan of the
TM00 resonance near the wavelength of $1546.3$ nm when the TM00 resonance near
1556 nm is selected for the auxiliary laser. Both modulation instability (MI)
and single-soliton states are observed for a pump power of 6 mW. (c) Recorded
data from the electrical spectrum analyzer (ESA) corresponding to the MI
(orange) and single-soliton (yellow) states. (d) Optical spectrum
corresponding to the single-soliton state with a measured FSR of $99.8$ GHz
(blue lines). The red dashed line is the simulated envelope of the single
soliton state based on the LLE method.

To generate soliton microcombs, we must overcome the thermo-optic bistability
that prevents adiabatic accessing of the soliton state. While there are
multiple approaches available, in this work we employ a cooling laser method
by introducing an auxiliary laser to mitigate the thermal effect [40]. In
general, the auxiliary laser is coupled to a resonant mode different from the
pump to avoid competing comb generation (either with a different radial order
or different polarization). Here, the competing Raman effect in 100-GHz-FSR
microrings can be used to our advantage, as we can employ the TM00 mode with
the suppressed Kerr effect for the thermal cooling. For example, by pumping
the TM00 resonance near $1546.3$ nm (for which the Kerr effect dominates)
while also coupling to the TM00 resonance near $1556$ nm (for which the Stokes
Raman dominates) in the counter-propagating direction (see the schematic in
Fig. 4(a)), we have successfully observed soliton step in the pump
transmission (Fig. 4(b)). The power of the auxiliary laser is typically
adjusted between one to three times of the pump laser to maximize the soliton
step. The dramatic change from the chaotic modulation-instability (MI) comb to
the phase-coherent soliton state is reflected in the photodetected signal when
it is connected to an electrical spectrum analyzer (Fig. 4(c)). Finally, the
optical spectrum of the soliton state is recorded by slowly tuning the pump
laser to the soliton step, confirming that it is indeed a single-soliton state
for pump powers as low as 6 mW (Fig. 4(d)). The experimental data also agrees
with the numerical simulation based on the LLE method, with the minor
deviation attributed to stronger resonator-waveguide coupling at longer
wavelengths (hence higher comb powers).

Figure 5: (b) Pump scan of the TE00 resonance near the wavelength of 1546nm
when the TM00 resonance near 1556 nm is selected for the auxiliary laser. A
single soliton step is observed for an input power around 12 mW. (b) Optical
spectrum corresponding to the single-soliton state. The red dashed line is the
simulated envelope of the single soliton state based on the LLE method.

For the TE00 mode, we carry out the same procedure by employing one of the
TM00 resonances with the suppressed Kerr effect for thermal cooling. As a
result, single soliton states can be adiabatically accessed for pump powers as
low as 12 mW (see Fig. 5). While its comb bandwidth is verified to be broader
than that of the TM00 mode, the average power of the comb lines for the TE00
mode is approximately 10 dB smaller. In addition, there are several avoided
mode crossings appearing in the comb spectrum, which further lower the comb
power and hinder the spectral expansion (see Fig. 5(b)).

Finally, it is worth comparing our results to Kerr soliton microcombs with
similar FSRs reported in other material platforms. As can be seen in Table 1,
the demonstrated soliton operation power (i.e., 6 mW) is among the lowest
reported numbers in the literature (despite having relatively low $Q$s when
compared to others). This is uniquely enabled by the stronger Kerr
nonlinearity offered by 4H-SiC, which is approximately 4 times of that of SiN
and 40 times of that of silica. In addition, we also note that our experiment
represents the first demonstration of 100-GHz-FSR soliton microcombs in any
SiC microresonators.

Material |  FSR (GHz) |  $\bm{Q}_{0}$ (M) |  Soliton pump power (mW) | Reference
---|---|---|---|---
Silica | $\bm{50.6}$ | $\approx 280$ | $\approx 3.0$ | S. Zhang et.al.[41]
LN | $\bm{44.84}$ | $\approx 4$ | 396 | Y. He et.al.[39]
Si3N4 | $\bm{99}$ | $\approx 15$ | $6.2$ | J. Liu et.al. [42]
$\bm{99.79}$ | 8-14 | 5.0-7.4 | O. B. Helgason et.al.[43]
4H-SiC | 350 | 5.6 | 2.3 | M. A. Guidry et.al. [36]
208 | 4.0 | $\approx 39$ | C. Wang et.al. [44]
99.8 | 4.4 (TM00) | 6.0 | This work
105.3 | 5.7 (TE00) | 12.0 | This work
Table 1: Selective list of Kerr soliton microcomb generation with microwave-
rate FSRs (i.e., $<=100$ GHz) in various material platforms. For comparison
within the 4H-SiCOI platform, We have also included experiments that have
reported soliton states in SiC regardless of the FSR choice.

##  6 Conclusion

In summary, we have demonstrated 100-GHz-FSR soliton microcombs in the
4H-SiCOI platform for the first time, which can be readily aligned to the DWDM
ITU grid spacing. Through careful dispersion engineering and optimized
nanofabrication, the on-chip pump power has been reduced to 6 mW while
supporting comb powers above -20 dBm per line near the pump wavelength. In
addition, our work has identified the potential limit from the Raman effect
and how to use it for adiabatic accessing of the soliton state. As such, we
believe the 4H-SiCOI platform can be highly competitive and promising for
supporting low-power soliton generation with properties tailored for narrow-
grid optical communications. Funding This work was supported by NSF (2131162)
and CableLabs University Outreach Program.

Acknowledgments The authors would like to thank Dr. Gregory Moille for helpful
discussions on the soliton generation. The authors acknowledge the use of
Bertucci Nanotechnology Laboratory at Carnegie Mellon University supported by
grant BNL-78657879 and the Materials Characterization Facility supported by
grant MCF-677785. R. Wang also acknowledges the support of Tan Endowed
Graduate Fellowship from CMU.

Disclosures The authors declare no conflicts of interest.

Data Availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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