Download Accelerating Structures: Resonant Cavities

Accelerator Laboratory:
RF ACCELERATING STRUCTURES
Summary
• Introduction
• Standing Wave Structures (Resonant cavities)
• Travelling Wave Structures (Iris loaded waveguides)
• Lab Experimental activity
 Characterization of an Aluminum model of the S-band RF GUN for the ELINP project
 Characterization of an Aluminum model of the S-band RF deflector for the
CTF3 project
The LNF RF Team:
D. Alesini, M. Bellaveglia, F. Cardelli, A. Gallo, A.Mostacci, L. Piersanti
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Fundamental relations of the relativistic dynamics
Rest
Energy
W0  m0 c
Relativistic
-factor
2
  v c,
  1 always !
Relativistic
-factor

Total Energy
Wk  W  W0 
1
1  2
  1 always !
m   m0
Kinetic Energy
W   m0c 2   W0
W 2  W02  p 2c 2
 (  1)m0c 2 
1
 m0 v 2 se   1
2
Newton’s 2nd Law
Lorentz Force
d
d
F  p  (mv )
dt
dt
F  q (E  v  B)
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Energy-velocity plot
Velocity variations are negligible at energies
well above the particle rest energy!
β= v/c
Wk[MeV]
e- relativistic (v  c) at W>1MeV (W0=511keV)
p relativistic at W>1000 MeV (W0=938MeV)
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Leptons (light particles) are pratically
fully relativistic in any existing dedicated
accelerators
(Wk>>W0,
with
the
exception of the very first acceleration
stage) while protons and ions are
typically weakly relativistic (Wk<W0 – but
not always, see high energy hadron
colliders such as the LHC).
For leptons the accelerating process
occurs at constant particle velocity (v ≈
c), while protons and ions velocity may
change a lot during acceleration. This
implies major important differences in
the technical characteristics of the
dedicated accelerating structures.
Particle energies are typically expressed in
electron-volt [eV], equal to the energy
gained by 1 electron accelerated through an
electrostatic potential of 1 volt:
1 eV=1.6x10-19 J
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Fundamental equation of the particle motion

  
dp
 q EvB
dt

ACCELERATION
Longitudinal Dynamics


p  momentum
m  mass

v  velocity
q  charge
BENDING AND FOCUSSING
Transverse Dynamics

B  magnetic field

E  electric field
Electric field
Deflection
(magnetic field)
Beam
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From Electrostatic to RF acceleration
The first historical particle accelerator was built by the Nobel prize Wilhelm Conrad Röntgen. It
consisted in a vacuum tube containing a cathode connected to the negative pole of a DC voltage
generator. Electrons emitted by the heated cathode were accelerated while flowing to another
electrode connected to the positive generator pole (anode). Collisions between energetic electrons
and anode produced X-rays.
The energy gained by the electrons travelling from cathode to
anode is equal to their charge multiplied the electrostatic
potential difference between the two electrodes.
X
cathode
anode
e- +
PRINCIPLE:
The DC voltage of a generator
is used to accelerate particles


dp
 qE
dt
W  qV
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Basic limitation: the energy gain
∆W=q∆V is proportional to the DC
voltage ∆V which is limited by
unavoidable breakdown phenomena!
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To increase the achievable maximum energy Van de
Graaff invented an electrostatic generator based on a
dielectric belt transporting positive charges to an
isolated electrode hosting an ion source. The positive
ions generated in a large positive potential were
accelerated toward ground by the static electric field.
DC voltage as large as 15 MV can be obtained (E ~ 15MeV)
APPLICATIONS
Still ~ 350 Van De Graaff are in operation
worldwide, typically at V<25MV, I<100mA. They
are used for:
Material analysis:
such as Semiconductors structure analysis,
X-ray production, …;
Material modification:
ion implantation for semiconductors
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RF Acceleration : the Wideröe “Drift Tube LINAC” (DTL)
Basic idea:
The particles produced by a
accelerated by the electric
gap between electrodes
alternatively to the poles
generator.
filament are
field in the
connected
of an AC
If the length of the tubes increases with the
particle velocity during the acceleration such
that the time of flight is kept constant and equal
to half of the RF period , the particles are
subject to a synchronous accelerating voltage
and experience an energy gain of ∆W=q∆V at
each gap crossing.
The original idea of Ising (1924) was
implemented by Wideroe (1927) who
applied a sine-wave voltage to a
sequence of drift tubes. The particles do
not experience any force while travelling
inside the tubes (equipotential regions)
and are accelerated across the gaps. This
kind of structure is called Drift Tube
LINAC (DTL).
gap
In principle a single AC voltage can be used to
indefinitely accelerate a beam, avoiding the
breakdown limitation affecting the electrostatic
accelerators.
Please notice that this technique requires
bunched beams, i.e. in order to be synchronous
with the external AC field, particles have to be
gathered in non-uniform temporal structures
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In 1928 Wideröe by means of 3 tubes (2 gaps) accelerated Na e K
ions at 50 keV with an RF of 25 kV and 1 MHz.
In 1931 Sloan e Lawrence used 30 tubes to accelerate Hg ions to
the energy of 1.25 MeV with an RF of 42 kV e 10 MHz; in 1934, by
using 36 tubes and a higher RF voltage, they got to 2.8 MeV.
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:
Beam -to-accelerating field synchronization
The
Wideröe
DTL
provides synchronization
between
a
charge
particle in motion and an
accelerating E-field.
If particles enter the LINAC with an energy Win, while travelling inside the n-th drift tube
they will show energy Wn and velocity vn given by:
Wn  Win  nqVkick
 n2  1
; vn c   n 

n
Wn
Ln TRF

vn
2
Ln 
The length of the n-th drift tube has to be

W0   1
Wn W0
2
1
1
vnTRF   n RF
2
2
The condition Ln<<RF (necessary to model the tube as an equipotential region)
requires <<1. The Wideröe technique can not be applied to relativistic particles
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Standing Wave (SW) Accelerating Structures: Resonant Cavities
High frequency accelerating fields synchronized with the beam motion are obtained by
exciting metallic structures properly designed. In this case the structure physical
dimensions are comparable with the e.m. field wavelength, and the actual spatial and
temporal field profiles have to be computed (analytically or numerically) by solving the
Maxwell equations with the proper boundary conditions.
Electric field
beam
Resonant cavities are (almost) closed volumes were the e.m fields can only exists in the
form of particular spatial conformations (resonant modes) whose components, including the
accelerating field Ez, rigidly oscillate at some specific frequencies  characteristic of the
mode (Standing Waves). The accelerating field along the beam path has the following form:
Ez (z,t)  Eˆ z (z) cos(t   )
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Resonant Cavity Shapes
Resonant cavities are (almost) closed volumes
/4
were the e.m fields can only exists in the form of
particular spatial conformations (resonant modes)
rigidly
oscillating
at
some
characteristics
frequencies (Standing Waves).
Reentrant or nose-cone cavities
/2
Disk-loaded or coaxial cavities
Pill-box or
cylindrical
cavities
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Resonant Cavity Examples
Cylindrical Cavity or Pill Box (TM010 Mode)
Coaxial Cavities
(TEM Mode)
Er  E  Hr  Hz  0
Ez (r)  E0 J0 ( p01r / a)
H (r)   j

E J ( p r / a)
 0 1 01
/4 resonators
Pill-box like (or bell-shape)
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Voltage
Current
/2 resonator
Acceleration with SW cavities
Let’s compute the energy gain W for a charge q crossing the
gap of an accelerating cavity at an almost constant velocity v.
Let’s also consider a symmetric cavity profile. We may write:
 z q  vt ;

 E z (r  0, z , t )  Eˆ z ( z ) cos( t   )
The Energy gain is obtained integrating the accelerating field
sampled by the charge along the trajectory, and may be
expressed as:
L/2
L/2
L / 2
L / 2
W   qE z (r  0, z, t  z / v)dz   qEˆ z ( z ) cos(z / v   )dz

cos( )

field long.symmetry sin( ) is an odd function
L/2
 qEˆ ( z) cos(z / v)dz
z
L / 2
For symmetric field profiles the maximum integrated accelerating voltage V is obtained when the
relative phase between field and particle is   0 . For asymmetric field profiles the maximum
accelerating voltage is obtained at values   0 , and it may be easily demonstrated that:
L/2
V
 Eˆ ( z ) e
z
j z v
dz
L / 2
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Cavity dissipation – Q factor
Real cavities are lossy. Surface currents dissipate energy, so that a certain amount of RF power must be
provided from the outside to keep the accelerating field at the desired level. If the external excitation is turned
off, fields inside the cavity decay exponentially with a time constant
 n characteristic of any given mode.
In frequency domain, the dissipation makes the modes resonating not
only at the mode natural frequency
width
n
around
n .
n
Both the bandwidth
time  n are related to the quality factor
Q  n
but in a frequency band of
Q
n
Vmax
Pgen
and the decay
of the mode defined as:
 n  2Q / n
U

P
n 3 dB  n Q
where U and P are the e.m. energy stored in the mode and the
corresponding power dissipation on the walls.
General expressions for U and P are:
1  2 1  2
U     E   H  d ;
4
4

Vol
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P
 0
1
1
2
Rs  H tan
d with Rs 

2 Surf .
 
2
conductivity skin depth
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Frequency and
material conductivity
Shunt impedance
One of the most important
parameter to characterize the
cavity accelerating modes is the
shunt impedance
R
2
V
1
R

2P
Rs
defined as:
 E z ( z ) e dz
2
H
d
tan

Surf .
j z / v
2
Mode field
configuration
The shunt impedance is the parameter that qualifies the efficiency of an accelerating mode. The
highest the value of R , the larger the attainable accelerating voltage for a given power expenditure.
Another very useful parameter is the ratio between shunt impedance R and quality factor Q :
2
P
1
R V

 
 Q  2 P U 2
2
Ez ( z) e

traject .
j z / v
dz
1  2 1  2 
Vol.  4  E  4  H  d
R Q  is a pure geometric qualification factor. In fact, for a given mode it is straightforward that
the  R Q  does not depend on the cavity wall conductivity, and its value is preserved if homothetic
The
expansions of a given geometry are considered.
The
R Q 
is a qualification parameter of the cavity geometrical design.
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Modes of a Resonant Cavity: General Problem
The resonant cavity modes are solutions of the
homogeneous Maxwell equations inside closed
volumes surrounded by perfectly conducting walls.
Homogeneous Maxwell Equations
(wave equation)
+
perfect metallic boundaries
The mathematical problem has the following formal
expression:
 2 
2  
 Wave equation
 E (r , t )   2 E (r , t )
t
  

  n  E (r , t )  0
Phasors
 
 Perfect boundary

   
 Field solenoidal ity
   E (r , t )  0

 
 
  2 E (r )  k 2 E (r )
   
n  E ( r )  0
  
   E (r )  0
According to the theory of linear operators, the solution is represented by a discrete set of eigen-functions
 
E n (r )
 
and their associated eigenvalues k n   n c . The magnetic field eigenfunctions Bn (r ) can be obtained from the

 
  
 
B
    E n (r )   j n Bn (r )
Maxwell 3rd equation:   E  
t phasors
 
The functions E n (r ) are the cavity modes, each one resonating at a certain specific frequency  n . The eigen 
 
functions are also a linear independent base, so that the actual fields E ( r , t ) , B ( r , t ) can always be represented as a
 
  j t  
  j t
linear superposition of the cavity modes: E ( r , t )   a n E n ( r ) e n ; B ( r , t )   a n Bn ( r ) e n
n
n
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Analytical field solutions: the Pill-box cavity
In the simplest cases the mode field configuration
can be calculated analytically, while in almost all practical
cases the solutions are computed numerically by means of
dedicated computer codes.
One of the most interesting didactical case is the cylindrical
or “pill-box” cavity. The pill-box cavity can be seen as a
piece of circular waveguide short-circuited at both ends by
metallic plates.
Wave Type
TM01
TM02
TM11
TE01
TE11
Circular waveguide modes
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Numerical Solutions
In the majority of cases analytical field solutions are
not available and numerical methods are applied.
There are various codes dedicated to the solution of
the Maxwell equations in closed and/or open volumes
starting from a discretized model of the structure under
study. Codes can be classified in various ways:
2D (2-dimensionals) and 3D (3-dimensionals) codes;
Finite differences and finite elements codes;
Time-domain and frequency domain codes.
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Grid
E
Dual grid
H
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Figure of merit of a SW cavity
DEFINITION
PARAMETER
Q-factor
Ratio between energy stored in the cavity and energy
dissipated in one cycle. It is also an indicator of the resonance
bandwidth.
1
4
Vol
2
1
4
 2

  2f ris ; U     E   H  dV ; P 
1
2
2
Rs  H tan
dS ; Rs 
2 Sup
0
Shunt Impedance R
Maximum accelerating voltage for a given dissipated RF
power.
Shunt Impedance per unit length Z
Maximum accelerating voltage for a given dissipated RF
power a per unit length.
Geometric factor R/Q
Geometric factor qualifying the accelerating cavity shape
regardless to frequency and wall conductivity.
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Q
U
P
V2
R
2P
Z
R
V2

2 L 2 PL
V2
V2
R Q

2 PQ 2U
FREQUENCY
SCALING
Q  1 R  f 1 2
s
R 1
 f 1 2
1
 f12
Rs L
Z
R
Rs
Q
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 f 0!!!
Input-Output coupling
Real cavities are never completely closed volumes. At the least,
apertures for beam transit (beam tubes) are required, as well as RF
input couplers to feed the cavity, and RF output couplers to probe the
field inside. The RF couplers can be of different types:
Magnetic (loop)
input coupler
Waveguide (slot)
input coupler
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 Electric couplers (Antennas): the inner of a
coaxial line connected from the outside
couples to the cavity mode E-field;
 Magnetic couplers (Loops): the cavity
mode B-field couples to a loop connecting
inner and outer conductors of a coaxial
line.
 Waveguide couplers: the cavity mode
fields are coupled to an external waveguide
of proper shape and cut-off through a hole
or a slot in the cavity walls
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Coupling and loading parameters of a cavity
The coupling strength of a port can be measured as the amount of power Pout extracted from the cavity
through the port itself for a given level of the mode fields inside. This leads to the definitions of the externalQ ( Qext ) (in analogy with the definition of the resonance quality factor Q ) and coupling coefficient  of a
coupler according to :
Qext  
U
;
Pout

Q0
P
 out
Qext Pwalls
where Q0 is the quality factor of the resonant mode, related only to the dissipation Pwalls on the cavity walls.
The extra-power flow through the cavity couplers, in addition to the power loss in the walls, may
significantly change the characteristics of the resonance. This effect is known as “cavity loading”. The
loaded cavity Q-factor QL is lowered by the power coupled out through the cavity ports and results to be:
QL 
U
PTot

U
Pwalls   Poutn
Poutn
1   n
Pwalls
1
1
1






QL  U
 U Q0
Qext n
Q0
It can be easily demonstrated that, provided that all the monitor
ports are weakly coupled (βn<<1), the input coupling of a cavity
is related to the input reflection coefficient ρ by:
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 in 
Q0
1 

Qextin 1  
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Tuning
The cavity resonant frequencies need to be continuously controlled during
operation. Actual frequencies are affected by thermal drifts and, in case of
superconducting cavities, by pressure variations in the cryogenic bath. Storage ring
cavities have to be largely detuned during beam injection and beam coasting to
compensate for beam loading variations.
The frequency control is normally obtained through small deformations of the cavity
boundaries. The Slater theorem can be used to compute the resonant frequency change,
according to:
   H


  H
2
V
2
0
V
  E 2 d
  E 2 d

U H  U E
U
Cavity tuning is normally actuated through:
 Cooling fluid temperature control (linac TW or SW
sections);
 Structure pushing/stretching by application of axial
forces (SC and multi-cell cavities);
 Variable penetration of tuning plungers in the cavities
volume (room-temperature, single cell cavities under
heavy beam-loading).
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Perturbations
The Slater theorem can be used also to measure the field profiles in the cavity by
inserting a perturbing object and measuring the frequency shifts while moving it
along selected paths.
The theorem mathematical expression is different in this case and form factors need
to be introduced because field lines are strongly deformed in this case, while they are
almost unperturbed when the boundary is slightly deformed (tuning case).

0

1
4U
 k
V
B||

 B||2  k B  B2  k E  E||2  k E  E2 d

It turns out that if the perturbing object is a perfectly
conducting sphere the values of the form factors are:
||

k E||  k E  3 ;
k B||  k B 
3
2
If a sphere of radius a is moved along the beam axis of a cavity, the E field profile and the
R/Q of a resonant accelerating mode can be estimated according to:
E2 

U
;
3
0   a
R
1

Q 2 0 a 3
2

 0 s ( z ) e j0 z / c dz
z
sign of the E-field profile
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Multi-cell Cavities
Multi-cell resonant cavities are very effective in
reducing the number of RF power sources and input
couplers.
The N-cell structure behaves like a system composed
by N oscillators coupled together.
According to the theory of coupled oscillators, each
single-cell mode degenerates in a set of N possible
coupled oscillation modes of the whole system,
characterized by a cell-to-cell phase advance given by:
 n 
n
N 1
n  0,1, ..., N  1.
SW multi-cell cavities are designed to use the last mode (  N 1   ) for acceleration. Provided
that Lcell   2 the beam and the RF are in-phase all along the structures, which brings to:
R  N Rcell ;
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Q  Qcell
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RF Superconductivity
Superconductivity is a very well-known
physical phenomenon that is widely
used in RF for particle accelerators.
Surface resistance is reduced by orders
of magnitude (5 typically) in
superconducting (SC) cavities, which
are then very suitable when large
gradients have to be sustained
continuously or in a high duty-cycle
regime.
RBCS vs.
T
(Log scale)
RBCS vs.
f
(Log-Log)
Surface resistance
John Bardeen, Leon Neil Cooper and John Robert
Schrieffer (BCS) , 1957.
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1.3 GHz, 2-cell cavity for Cornell ERL injector.
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Examples of multi-cell cavity: the TESLA / ILC cavity
In the last 25 years superconducting
cavities have become technological
masterpieces providing incomparable
performances. Superconductivity applied
to RF resonant cavities allows gaining
about 5 orders of magnitude in Q factors!
A noticeable example is the 9-cells,
1.3 GHz cavity originally designed for the
DESY TeV linear collider proposal
TESLA, and presently adopted as the
baseline for existing (FLASH , XFEL DESY), and future (International Linear
Collider - ILC) projects.
The most widely used material for superconducting RF
is Niobium (Nb), usually bulk but also as film on a
metallic substrate. Lead (Pb) is also used.
The SC cavity geometries have to be very simple and
smooth, since any tip or corner is a potential site for
local hot spots which may cause quenches (i.e. loss of
the superconducting status of the material).
Moreover, the whole internal surface of the structure
must be easily accessible to allow effective cleaning
and polishing treatments.
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Traveling Wave (TW) Accelerating Structures
Particle beams can be accelerated not only by standing waves but also by traveling waves.
In this case it is necessary to let an e.m. wave with non-zero longitudinal electric field travel together with
the beam in a special guide in which the wave phase velocity matches the particle velocity. If this is the case
the beam absorbs energy from the wave and it is continuously accelerated.
Constant cross-section waveguides
By solving the wave (Helmholtz) equation it turns out that an e.m. wave propagating in a constant cross
section guide will never be synchronous with a particle beam since the propagation speed (i.e. the wave
phase velocity) is always larger than the speed of light c.
Let’s consider for instance the first mode (TM01) with a non zero longitudinal electric field of a circular
waveguide . The accelerating field at a given frequency  has the following expression:
propagation constant
phase velocity
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The wave phase velocity and the propagation constant  are simply related by:
while, solving the wave equation, the propagation constant and the phase velocity result to be
functions of the frequency according to:
specific of the guide
geometry and of the
selected mode
As already mentioned the wave phase velocity is always larger than c, so that the
accelerating field and the particles in the beam can never be synchronous.
Please notice that the phase velocity is not the energy propagation velocity of the structure. In
fact the energy propagates with the group velocity given by:
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The  vs.  plot is called dispersion curve of the
waveguide.
Noticeable aspects are:
Dispersion plot
a) the tangent of the ph angle is the wave phase
velocity at the operating frequency  ;
b) the propagation constant  is real (propagating
mode ) only if  > c ( c = cut-off frequency of the
selected mode);
*
c) if  < c the mode does not propagate in the
structure;
d) the tangent of the g is the wave group velocity in
the structure. In constant cross section waveguide,
phase and group velocities have the same sign
(forward waves), but this is not true in general since in
some peculiar guides (like periodical iris loaded
structures) they may have opposite signs (backward
waves);
e) The plot for negative  corresponds to waves
propagating in opposite direction (towards negative z).
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*
Constant cross section
waveguides:
 c  k
2
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2
t
Periodic structures
In order to slow-down the wave phase velocity, irisloaded periodic structure are used.
According to the Floquet theorem, the field in this kind
of structures is that of a special wave travelling within a
spatial periodic profile, with the same spatial period D of
the structure. The periodic field profile can be Fourier
expanded in a series of traveling waves (spatial
harmonics) with different phase velocity according to:
z
a)
b)
c)
d)
e)
A typical dispersion curve of an iris loaded structure has the following
characteristics:
the plot is periodic respect to the variable , and the period is 2/D;
each period is the dispersion curve of a different spatial harmonic;
the geometry of the guide can be designed such that the fundamental
spatial harmonic E0 is synchronous with the beam (i.e. phase velocity =
beam particle velocity) for a selected operating frequency *;
the high-order harmonics (n=1,2,3,...) are asynchronous respect to the
beam, so they do not contribute to the acceleration;
periodic structures can only operate in limited frequency bands
(stopbands associated with periodicity, as in other physics process ...)
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RF Accelerating Structures
Merit Figures of a periodic structure
Traveling wave (TW) accelerating structures are typically qualified by the following merit
figures :
a) Shunt impedance per unit length Z: It is defined as the ratio between the squared amplitude
of the fundamental harmonic accelerating field (E0) and the power dissipated per unit length
in the structure (dP/dz):
2
𝐸0
𝑍=
÷ 𝜔1
𝑑𝑃 𝑑𝑧
2
;
𝑍 = Ω/𝑚
The higher the Z value, the higher the available accelerating field for a given RF power
dissipation per unit length;
b) Q factor per unit length: It is defined as the ratio between the stored energy per unit
length (w) the power dissipated per unit length (dP/dz), times the operating frequency *:
𝑄=
𝜔∗
𝑤
÷ 𝜔 −1
𝑑𝑃 𝑑𝑧
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2
;
𝑄 = 𝑑𝑖𝑚𝑒𝑛𝑠𝑖𝑜𝑛𝑙𝑒𝑠𝑠
RF Accelerating Structures
c) Z/Q ratio: From previous definitions it turns out:
Similarly to the SW structures, this parameter only depends on the structure
design, which means that it is independent on the conductivity of the structure
inner surface. Since for a given E0 value the stored energy per unit length w is
inversely proportional to 2 (for homotetic scaling of the dimensions):
and since the operating frequency * is inversely proportional to the dimension scaling
factor, the Z/Q factor, differently from the SW case, results to grow linearly with frequency.
d) Group velocity vg: The group velocity is defined as:
and it represents the velocity of the energy flow in the structure. It may be
demonstrated that the group velocity approximately scales with a and b
dimensions according to:
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RF Accelerating Structures
The group velocity vg is a very important qualifying parameter of a TW
structure. The filling time f, i.e. the time necessary to propagate an RF
wavefront from the input to the end of a TW accelerating section of length L is
given by:
High group velocities allow reducing the duration of the RF pulse feeding the
structure. However, the RF power P flowing through the structure (P = flux of
the Poynting vector) and the stored energy per unit length w are related by:
and since E0 is proportional to the square root of w, clearly a low group velocity is preferable
to increase the effective accelerating field for a given power flowing in the structure.
e) Frequency *. Since:
To increase Z the operating design frequency must be the highest possible. However, the
power available from ordinary RF power sources at high frequencies decreases rapidly with
frequency. Also RF power density on the structure surface increases and the irises diameter
decrease linearly with frequency, so problems related with heating increase and beam stay
clear reduction at high frequencies have to be taken into account.
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RF Accelerating Structures
f) attenuation:. Because of the wall dissipation, the RF power flux
decreases along the structure according to the following equation:
where  is the structure attenuation coefficient. It may be shown that:
𝑑𝑃
= −𝛼𝑃
𝑑𝑧
𝜔
𝛼=
÷ 𝜔3
𝑄𝑣𝑔
2
In a purely periodic guide, made by a sequence of identical cells (also called “constant
impedance structure”),  does not depend on z and both the RF power flux and the intensity
of the accelerating field decay exponentially along the structure :
𝑃 𝑧 = 𝑃𝑖𝑛 𝑒 −𝛼𝑧 ; 𝐸0 𝑧 = 𝐸0𝑖𝑛 𝑒 −𝛼𝑧
2
It is possible to design structures with nearly constant accelerating field along z . The
diameter of the cell irises has to be gradually reduced, reducing the group velocity along the
structure in order to keep nearly constant the energy stored per unit length w and power
dissipated per unit length dP/dz . These are called “constant gradient structures”, and are not
exactly periodic.
g) Working mode: It is defined as the phase advance of the
fundamental harmonic over a period D:
For practical reasons related to the experimental
measurement of the dispersion curve of actual devices,
the working modes are typically designed to be:
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RF Accelerating Structures
Examples of TW structure: the Stanford LINAC (SLAC)
This is the most widely diffused TW iris-loaded accelerating
structure, developed at the "Stanford Linear Accelerator Center"
(SLAC).
It works in S-band (f=2856 MHz) and it is composed by 86
accelerating cells working in 2/3 mode. It has been produced in
various types ("constant impedance" and "constant gradient") and
generations, and tested and operated up to 30 MV/m and beyond.
L≈3m
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RF Accelerating Structures
EDIT 2015 – Accelerator Lab
RF Measurements
• Instrumentation Description
• Experience #1: characterization of a SW cavity
(S-band RF Gun)
• Experience #2: characterization of a TW
structure (S-band RF deflector)
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RF Accelerating Structures
VNA conceptual scheme: the 4 blocks
1
Source
2
Signal
Separation
Incident Reflected
3
Z0 LOAD
DUT
Transmitted
Detector and
Receiver
4
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Process and
Display
A Vector Network Analyzer
measures the scattering matrix
elements sij – i.e. Input/output
matching and forward/backward
transfer function – of a Device
Under Test.
An internal sine-wave
sorce is made sweeping a
given frequancy span and
reflected and transmitted
signas are measuerd and
normalized to the source.
The direction of the
excitation
can
be
inverted.
RF Accelerating Structures
Network Analyzer Block Diagram
Incident
Transmitted
DUT
SOURCE
Reflected
SIGNAL
SEPARATION
INCIDENT
(R)
REFLECTED
(A)
TRANSMITTED
(B)
RECEIVER / DETECTOR
PROCESSOR / DISPLAY
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RF Accelerating Structures
Spectrum Analyzer Block Diagram
A
Spectrum
Analyzer
characterizes signals in
frequency domain. An
internal source sweeps a
selected frequency range
to downconvert and filter
the input signal. The IF
filter output power reveals
the spectral content of the
signal at the instantaneous
frequency scanned by the
instrument.
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RF Accelerating Structures
Digital Oscilloscopes
A Digital Oscilloscope characterizes signals in time domain. The input signal is buffered by a
front-end amplifier and then A-to-D converted. Instrument BW and resolution depend
essentially by the characteristics of the front-end ADC . Digital data stream is stored,
processed and displayed according to experimental needs. Special algorithms working on the
acquired samples allow smart triggering of the instrument.
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RF Accelerating Structures
Experience #1: ELI-NP S-band RF-Gun
Design
value (Cu)
Meas. value
(Al)
Working frequency
2.856 (GHz)
?
Working mode
π
?
Unloaded Q
14500
?
Parameter
Input coupling coeff. (ß) 3
?
Input external Q
4800
?
Filling time
420 ns
?
Shunt impedance (R)
1.7 MΩ
?
R/Q factor
117 Ω
?
Repetition rate
100 Hz
X
Max RF input power
16 MW
X
Peak field at cathode
120 MV/m
X
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cathode
bead
½ cell cell
beam pipe
RF Accelerating Structures
Experience #1: RF characterization of an
Aluminum model of the ELI-NP S-band RF-Gun
• f0, fπ, ∆f, QL, Q0, ß measurement with Network Analizer
• Longitudinal electric field profile measurement with “bead-drop”
technique (mode π):
− Ticks drawn on the fishing line are equally spaced every 5 mm
− Fill an excel spreadsheet with 2 columns: bead position (z) and fres
− Calculate and plot ∆f/f vs z;
• Longitudinal electric field profile measurement with “bead-drop”
technique (mode 0):
− Same as mode π… but centered at f0
• Shunt impedance (or R/Q) calculation:
− From the same spreadsheet, calculate Vacc and then use the formula
for R (or R/Q);
• Filling time measurement (time domain with realistic RF pulse)
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RF Accelerating Structures
Travelling Wave structure
The Aluminum model of the CTF3
combiner ring RF deflector
accelerating cell
nom. Energy En [MeV]
184
max. Energy Emax [MeV]
350
frequency f [MHz]
number of cells N c
2998.55
10
2π/3
De-phasing/cell
total length L [cm]
group velocity v g / c
33
-.0244
phase velocity v ph / c
filling time 
F0
Deflection

1
[ns]
Shunt impedance Rs  (E )
[mrad]
max RF Power PRF [MW]
beam pipe
RF power
couplers
46
2
2PRF [MΩ]
0.19
5
8
Dispersion curve
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RF Accelerating Structures
Experience #2: RF Travelling Wave structure
• Example of a disassembled C-band TW structure;
• BW measurement at fRF=3 GHz with Network Analizer;
• Filling time measurement #1:
• Using a realistic pulse from RF signal generator (fRF=3 GHz, tpulse =1
µs, rep.rate=10-100 Hz) fed to the structure;
• Filling time measurement #2:
• Using a CW RF reference with amplitude modulation, according to
the scheme of slide 3 (fmod=5 MHz, Ampl=1 Vpp, offset=2.5 V);
• With spectrum analizer monitor the frequency content of the feeding
signal;
• Measure with the oscilloscope the time needed to traverse the
structure from the distance of the 2 maxima.
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RF Accelerating Structures
TW structure filling time measurement setup
fRF
RF
LO
CPL
IF
TW structure
CPL
ATT 10 dB
Funct. generator
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TTL
TRG
Oscilloscope
RF Accelerating Structures
N.A. settings and measurement results
Parameter
Mode π
Mode 0
Working
frequency
2.85564 (GHz)
2.81422 (GHz)
S11 @ fres
-10.8 dB
(lin = 0.289)
-23.2 dB
(lin = 0.07)
Q0
8990
10100
QL
3200
4700
Coupling
coefficient (ß)
1.81
1.15
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General measurement
settings:
- IF BW 40 kHz
- # points 1601
- fcenter = 2.83 GHz
- fspan = 100 MHz
Single mode measurement
settings:
- IF BW 40 kHz
- # points 1601
- fcenter = fres
- fspan = 10 MHz
RF Accelerating Structures