Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010
Lecture 5
Wave guides and Resonators
A.Nassiri
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 2
Dispersion
First, consider a very simple sinusoidal electromagnetic wave propagating along along the x-axis, with electric field
( )kxtAE −ω= cos
kv
vx
tAE
p
p
ω=
−ω= cos
vp is the phase velocity. The phase velocity is the velocity at which the crest of the sinusoidal wave move through space. If one moves the observation point from 0 to x, the wave will arrive
Later and tp is referred to as the phase delay.p
p vx
t =
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 3
Dispersion
In vacuum (free space) the wave vector is given by k = ω/c and the phase velocity is just the vacuum velocity of light. In a material the wave vector is given by k = ωn/c so that the phase velocity is the vacuum velocity of light divided by the refractive index n.
Lets consider the a superposition of two waves at slightly different frequency, that is:
( ) ( )( ) ( )kxtkxtA
xktAxktAE
∆−ω∆−ω=−ω+−ω=
coscoscoscos
22211
I have defined four new quantities as:
kkkkkk
∆−=ω∆−ω=ω∆+=ω∆+ω=ω
22
11 ,,
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 4
Dispersion
0
2
→∆∂ω∂
→∆
ω∆=
−ω∆
−ω=
kkk
v
where
vx
tvx
tAE
g
gp
for
coscos
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 5
Dispersion
vg is the group velocity and vp is the phase velocity. The crests of the wave still move at the phase velocity but the modulations move at the group velocity. As before the group delay is defined by: tg = x/vg
There is another, slightly different, definition of the phase andgroup delay. When a wave travels over a certain distance x, throughsome medium, it will accumulate phase. In complex notation, theoutput field is related by the input field by
kx
eEE jinout
=φ= φ
Referring to these equations, it can be seen that the phase andgroup delay can also be expressed as:
ω∂φ∂
=ωφ
= gp tt ,
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 6
Dispersion
How fast do signals travel?
When an electromagnetic wave travels through a medium, itaccumulates phase. Naturally, the question arises what the velocityis at which a signal propagates. This in not the phase velocity. Thephase velocity is the velocity of the waves or, put in other words, itis the velocity at which the zero crossings of the field propagate.Thus, the phase velocity at frequency ω0 is the velocity at whichthe wave E(t,x)∝ cos(ω0t-kx) propagates. A perfect cosine repeatsitself every 2π radians and cannot therefore transport anyinformation. Information is, for example, a series of bits that canonly be encoded on a electromagnetic wave by modulating it. Themodulations propagates with the group velocity. Right ?
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 7
Dispersion
Unfortunately, things are not that simple. Consider a wave guide, consisting of a hollow metal tube filled with air.
Ld
It can be shown that an electromagnetic wave traveling through this wave guide will accumulate the phase
. Assumingthat the electric field vector iseverywhere perpendicular to the metal-air interface (TE mode).
. xc=1.841.Using φ=kL,we can write:
( ) 12
−
ωωω
=ωφc
cc
L
dcxc
c2
=ω
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 8
Dispersion
( ) 22cck ω−=ω
0 10
1
( )12102 ×πω
( )12102 ×πck
Cut-off
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 9
Dispersion
The frequency scale ~1THz (λ~1mm)
The “light line” corresponds to light propagating in free space
At very large k, the dispersion curve becomes the light line
Thus, the wavelength of the light (k=2π/λ) becomes much shorter than the diameter of the wave guide.
When the wavelength becomes of the order of the diameter of the wave guide (k→0), the dispersion curve flattens. At k=0, the frequency has finite value, therefore the phase velocity, which is ω/k, is infinite! An infinite velocity means the electromagnetic wave travels the distance L through the wave guide in zero time.
On the other hand, the group velocity, given by is zero at k=0!Thus, in a wave guide at cut-off the phase velocity is infinite and the group velocity is zero. If signals travel with the group velocity then at cut-off signals do not travel at all. This a good example where the group velocity does not represent the signal velocity.
kv g ∂
ω∂=
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 10
Dispersion
What happens to electromagnetic waves that have frequencies below cutoff frequency?These waves are in a “forbidden region,” much like the bad gap in a semiconductor.
( ) 12
−
ωωω
=ωφc
cc
L ( )2
1
ωω
−ω
=ωc
cc
ik
Therefore, below cutoff the electromagnetic wave is no longer a propagating wave with well defined wavelength. Instead, the wave decays exponentially and smoothly. Such wave is called evanescent wave. Sine k is imaginary, ik is a purely real number.
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 11
Dispersion for Waveguide
βz
ω
ωc
Slope = group velocity = vg
Parallel Plate Wave guide
Slope=vpz
βz1 βz2 βz3
ω1
ω2
ω3
} Slope=vpz
µε=
1Slope
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 12
The cavity resonator is obtained from a section of rectangular wave guide, closed by two additional metal plates. We assume again perfectly conducting walls and loss-less dielectric.
d
bdpβ
bnβ
amβ
z
y
x
π=
π=
π=
Rectangular Waveguide Resonator (Recap)
a
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The addition of a new set of plates introduces a condition forstanding waves in the z-direction which leads to the definition of oscillation frequencies
222
21
+
+
µε=
dp
bn
am
fc
The high-pass behavior of the rectangular wave guide is modified into a very narrow pass-band behavior, since cut-off frequencies of the wave guide are transformed into oscillation frequencies of the resonator.
Rectangular Waveguide Resonator (Recap)
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 14
In the wave guide, each mode isassociated with a band of frequencieslarger than the cut-off frequency.
In the resonator, resonant modes canonly exist in correspondence of discreteresonance frequencies.
0 fc1 fc2 f fc10 fc2 f
Rectangular Waveguide Resonator (Recap)
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 15
The cavity resonator will have modes indicated as
TEmnp TMmnp
The values of the index corresponds to periodicity (number of sine orcosine waves) in three direction. Using z-direction as the reference forthe definition of transverse electric or magnetic fields, the allowedindices are
===
===
,...,,,,...,,,,...,,,
,...,,,,...,,,,...,,,
321032103210
321032103210
pnm
TMpnm
TE
With only one zero index m or n allowed
The mode with lowest resonance frequency is called dominant mode. In case a ≥ d>b the dominant mode is the TE101.
Rectangular Waveguide Resonator (Recap)
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 16
Note that a TM cavity mode, with magnetic field transverse to the z-direction, it is possible to have the third index equal zero. This is because themagnetic field is going to be parallel to the third set of plates, and it cantherefore be uniform in the third direction, with no periodicity.
The electric field components will have the following form that satisfies theboundary conditions for perfectly conducting walls.
π
π
π
= zdp
ybn
xa
mE xx sinsincosE
π
π
π
= zdp
ybn
xa
mE yy sincossinE
π
π
π
= zdp
ybn
xa
mE zy cossinsinE
Rectangular Waveguide Resonator (Recap)
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 17
The amplitudes of the electric field components also must satisfy the divergence condition which, in absence of charge is
00 =
π
+
π
+
π
⇒=⋅∇ zyx Edp
Ebn
Ea
mE
The magnetic field intensities are obtained from Ampere’s law:
π
π
π
ωµβ−β
= zdp
ybn
xa
mj
EEH zyyz
x coscossin
π
π
π
ωµβ−β
= zdp
ybn
xa
mj
EEH xzzx
y cossincos
π
π
π
ωµβ−β
= zdp
ybn
xa
mj
EEH yxxy
z sincoscos
Rectangular Waveguide Resonator (Recap)
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 18
Similar considerations for modes and indices can be made if the other axesare used as a reference for the transverse field, leading to analogousresonant field configurations.
A cavity resonator can be coupled to a wave guide through a small opening.When the input frequency resonates with the cavity, electromagneticradiation enters the resonator and a lowering in the output is detected. Byusing carefully tuned cavities, this scheme can be used for frequencymeasurements.
Rectangular Waveguide Resonator (Recap)
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 19
Example of resonant cavity excited by using coaxial cables.
The termination of the inner conductor of the cable acts like an elementary dipole (left) or an elementary loop (right) antenna.
Excitation with a dipole antenna Excitation with a loop antenna
Rectangular Waveguide Resonator (Recap)
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 20
zjcy
zjcx
zjz
z
z
z
z
eybn
xa
mA
bm
jE
eybn
xa
mA
an
jE
eybn
xa
mAH
E
β
β
β
π
ππ
ωµπ
λ−=
π
ππ
ωµπ
λ=
π
π
=
=
cossin
sincos
coscos
2
2
2
2
4
4
0
Field Expressions For TE Modes – Rec. WG
g
yx
EH
η=
g
xy
EH
η±=
m,n = 0,1,2,… but not both zero
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 21
22
21
+
µε=
bn
am
fc 222
+
=λ
bn
am
c
( ) ( )22 11 ffcc
g−
λ=
λλ−
λ=λ
( ) ( )22 1
1
1
1
ccpz
ff λλ−µε=
−µε=ν
( ) ( )22 11 ccg
ff λλ−
εµ=
−
εµ=η
Field Expressions For TE Modes – Rec. WG
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 22
zj
g
cy
zj
g
cx
zjz
z
z
z
z
eybn
xa
mA
bn
jE
eybn
xa
mA
am
jE
eybn
xa
mAE
H
β
β
β
π
ππ
πλλ
=
π
ππ
πλλ
=
π
π
=
=
cossin
sincos
coscos
2
2
0
2
2
g
yx
EH
η=
g
xy
EH
η±=
m,n = 1,2,3,…
Field Expressions For TE Modes – Rec. WG
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 23
22
21
+
µε=
bn
am
fc 222
+
=λ
bn
am
c
( ) ( )22 11 ffcc
g−
λ=
λλ−
λ=λ
( ) ( )22 1
1
1
1
ccpz
ff λλ−µε=
−µε=ν
( ) ( )22 11 ccg ff λλ−εµ
=−εµ
=η
Field Expressions For TE Modes – Rec. WG
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Determine the lowest four cutoff frequencies of the dominant mode for three cases of rectangular wave guide dimensions b/a=1,b/a=1/2, and b/a =1/3. Given a=3 cm, determine the propagating mode(s) for f=9 GHz for each of the three cases.
The expression for the cutoff wavelength for the TEm n mode where m=0,1,2,3,.. and n=0,1,2,3,.. But not both m and n equal to zero and for TMmn mode where m=1,2,3,.. And n=1,2,3,.. is given by
22
22
1
+
=λ
bn
am
c
The corresponding expression for the cutoff frequency is 2
222
21
221
+
µε=
+
µε=
λν
=ba
nmab
na
mf
c
pc
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 25
The cutoff frequency of the dominant mode TE10 is . Hence µεa21
[ ]2
2
10
+=
ba
nmf
f
TEc
c
TE10
TE01
TM11
TE11
TE20
TE02
TM21
TM12
TE21
TE12
[ ]10TEc
cf
f1=ab
1 2 3 4 5 2 5
21
=ab
TE10
TE01
TE20
TE11
TM11
TE21
TM21
[ ]10TEc
cf
f1 2 3 4 5 5 8
31
=ab
TE10 TE20
TE30
TE01
TM11
TE11
[ ]10TEc
cf
f
1 2 3 4 5 10
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 26
Rectangular Cavity Resonator
Add two perfectly conducting walls in z-plane separated by a distance d. For B.C’s to be satisfied, d must be equal to an integer multiple of λg/2 from the wall. Such structure is known as a cavity resonator and is the counterpart of the low-frequency lumped parameter resonant circuit at microwave frequencies, since it supports oscillations at frequencies foe which the foregoing condition, that is
d=l λg/2, l=1,2,3,… is satisfied.
a
bdx
y z
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 27
Hence for a signal of frequency f=9GHz, all the modes for which is less that 1.8 propagate. These modes are:
TE10,TE01,TM11,TE11 for b/a=1
TE10 for b/a=1/2
TE10 for b/a=1/3
[ ]10TEc
cf
f
So for b/a≤1/2, the second lowest cutoff frequency which corresponds to that of the TE20 mode is twice of the cutoff frequency of the dominant TE10 . For this reason, the dimension b of the a rectangular wave guide is generally chosen to be less than or equal to a/2 in order to achieve single-mode transmission over a complete octave( factor of two) range of frequencies.
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 28
Rectangular Cavity Resonator
Substituting for and rearranging, we obtainλg
( )2
22
2
211
1
2
=
λ−
λ
λλ−
λ=
dp
pd
c
c
Substituting for givesλc222
2 2221
+
+
=
λ dp
bn
am 222
222
1
+
+
=λ
dp
bn
am
222
2221
+
+
µε=
λν
=dp
bn
am
f posc
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 29
Quality Factor Q
The quality factor is in general a measure of the ability of a resonator to store energy in relation to time-average power dissipation. Specifically, the Q of a resonator is defined as
wall
strPWQ ω=π=
cycleper dissipatedEnergy storedenergy Maximum2
mestr WWW +=Consider the TE101 mode:
+
µ=
π
π
πωµε
=ε
= ∫∫∫∫
116
44
2
22
2
0
2
00
222
daHabdW
dxdydzdzsin
axsinHadvEW
e
abd
v ye
2 and 11
02
22
22101
2 adxaxsin
da
a=
π
+
µεπ
=ω=ω ∫
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 30
The time average stored magnetic energy can be found as
dxdydzdz
ax
dz
ax
d
aH
dvHHW
abd
zv xm
π
π
+
π
πµ
=
+µ
=
∫∫∫
∫
222
0
22
2
00
2
22
4
4
sincoscossin
+
µ= 1
16 2
22
daHabd
Wm
Note that the the time-average electric and magnetic energies are preciselyequal. This should be true in general simply follows from the complex Poyting’stheorem. Physically, the fact that energy cycles between being purely electric,partly electric and partly magnetic, and purely magnetic storage, such that on theaverage over a period, it is shared equally between the electric and magneticforms. The total time-average stored energy is
+
µ=+= 1
8 2
22
daHabd
WWW mestr
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 31
We now need to evaluate the power dissipated in the cavity walls. This dissipation will be due to the surface currents on each of the six walls as induced by the tangential magnetic fields, that is . Note that the
power dissipation is given by and that
is the surface resistance.
HnJ s ×= ˆ
ss RJ 2
21
tanHJ s =
σfR ms µπ=
+++
==
∫∫∫∫∫∫
∫
==
bottomtop
a
xz
d
leftright
b
xz
d
front,back
a
zx
bs
wallswall
dxdzHHdydzHdxdyHR
dsHR
P
,,
tan
0
22
00
20
00
20
0
2
2222
2
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 32
After completing the integration steps, we obtain:
+
+
+
= 121
4 3
3
2
222
d
adb
d
adadHR
P swall
Therefore the quality factor Q, is
+
+
+
+
πµ=
ω=
121
1
3
3
2
2
2
2
101
d
adb
d
ada
d
a
DRabf
PW
Qswall
str
Substituting for f101, gives
+
+
+
+
ηπ=
121
1
23
3
2
2
23
2
2
2
d
adb
d
ada
d
a
dRb
Qs
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 33
For a cubical resonator with a = b = d, we have
( )
( )[ ]21101
221011
101
33
112
121
−
−
σµπ=δδµ
µ=
πµ=
+
µε=µε=
mms
cube fa
Raf
Q
dafaf
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 34
Air-filled cubical cavity
We consider an air-filled cubical cavity designed to be resonant in TE101 mode at 10 GHz (free space wavelength λ=3cm)with silver-plated surfaces (σ=6.14×107S-m-1, µm= µ0.. Find the quality factor.
( ) cmf
cf
aaf 122222
1121101101
1101 .≈
λ==
εµ=⇒µε= −
At 10GHz, the skin depth for the silver is given by
( ) mµ≈×××π×××π=δ−− 6420101461041010
21779 ..
and the quality factor is
0001164203
1223
,.
.≅
µ×≅
δ=
mcma
Q
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 35
Previous example showed that very large quality factors can be achieved with normal conducting metallic resonant cavities. The Q evaluated for a cubical cavity is in fact representative of cavities of other simple shapes. Slightly higher Q values may be possible in resonators with other simple shapes, such as an elongated cylinder or a sphere, but the Q values are generally on the order of magnitude of the volume-to-surface ratio divided by the skin depth.
( )
cavity
cavity
s
ts
v
wall
m
wall
strS
V
dsHR
dvHf
PW
PW
Qδ
≅
µπ
=ω
=ω=
∫∫ 2
2
22
22
2
Where Scavityis the cavity surface enclosing the cavity volume Vcavity.
Although very large Q values are possible in cavity resonators,disturbances caused by the coupling system (loop or aperture coupling),surface irregularities, and other perturbations (e.g. dents on the walls) inpractice act to increase losses and reduce Q.
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 36
Dielectric losses and radiation losses from small holes may be especially importantin reducing Q. The resonant frequency of a cavity may also vary due to thepresence of a coupling connection. It may also vary with changing temperature dueto dimensional variations (as determined by the thermal expansion coefficient). Inaddition, for an air-filled cavity, if the cavity is not sealed, there are changes in theresonant frequency because of the varying dielectric constant of air with changingtemperature and humidity.Additional losses in a cavity occur due to the fact that at microwave frequencies forwhich resonant cavities are used most dielectrics have a complex dielectricconstant . A dielectric material with complex permittivity draws aneffective current , leading to losses that occur effectively due to
ε ′′−ε′=ε jEJeff ε ′′ω=
*effJE ⋅
The power dissipated in the dielectric filling is
dydxdzE
dvEEdvJEP
a b dy
vv
effdielectric
2
0 0 02
21
21
∫ ∫ ∫
∫∫ε′′ω
=
ε ′′ω⋅=⋅=
**
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 37
Using the expression for Ey for the TE101 mode, we have
ε ′′ε′
=ω=
+
µω
ε′ε ′′
=
d
strd
dielectric
PW
Q
d
aabdHP
1
8 2
22
dvEPand
dvEWW
vydielectric
vymstr
2
2
2
22
∫∫
ε′′ω=
ε′==
The total quality factor due todielectric losses is
cd QQQ111
+=
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 38
Teflon-filled cavity
We found that an air-filled cubical shape cavity resonating at 10 GHz has a Qc of11,000, for silver-plated walls. Now consider a Teflon-filled cavity, with ε= ε0(2.05-j0.0006). Find the total quality factor Q of this cavity.
[ ]rrr
da fc
aa
c
aff
ε′=⇒
εµ=
ε′µ≅= =
22
22
12101
µr=1 for Teflon. This shows that the the cavity is smaller, or a=b=d=1.48cm. Thus we have
Or times lower than that of the air-filled cavity. The quality factor Qd dueto the dielectric losses is given by
rε′
76843
≅δ
=a
Qc
rε′
2365≅+
=cd
cdQQ
QQQ
Thus, the presence of the Teflon dielectric substantially reduces the quality factorof the resonator.
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 39
Cylindrical Wave Functions
x
y
z ρ
φ
z
The Helmholtz equation in cylindrical coordinates is
011 22
2
2
2
2 =ψ+∂
ψ∂+
φ∂
ψ∂
ρ+
ρ∂ψ∂
ρρ∂∂
ρk
z
The method of separation of variables gives the solution of the form
0111 22
2
2
2
2 =+∂
+φ∂
Φ
Φρ+
ρρ
ρρk
z
ZdZ
dddR
dd
R
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 40
Cylindrical Wave Functions
22
21zk
z
ZdZ
−=∂
( ) 01 2222
2=ρ−+
φ∂
ΦΦ
+ρ
ρρ
ρzkk
dddR
dd
R
22
21n
d−=
φ∂
ΦΦ
( ) 02222 =ρ−+−ρ
ρρ
ρzkkn
ddR
dd
R
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 41
( )[ ]
0
0
0
22
2
22
2
22
222
=+∂
=Φ+φ∂
ψ
=−ρ+ρ
ρρ
ρ
=+
ρ
ρ
ZKz
Zd
nd
RkkddR
dd
R
kkk
k
z
z
z
p satisfy toDefine
Cylindrical Wave Functions
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 42
These are harmonic equations. Any solution to the harmonic equation we call harmonic functions and here is denoted by h(nφ)and h(kzz). Commonly used cylindrical harmonic functions are:
Where is the Bessel function of the first kind,
Is the Bessel function of the second kind, is the Hankel
function of the first kind, and is the Hankel function of the second kind.
( ) ( ) ( ) ( ) ( )ρρρρρ ρρρρρ kHkHkNkJkB nnnnn21 ,,,~
( )ρρkJ n ( )ρρkN n
( )ρρkH n1
( )ρρkH n2
Cylindrical Wave Functions
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 43
Any two of these are linearly independent.
A constant times a harmonic function is still a harmonic function
Sum of harmonic functions is still a harmonic function
We can write the solution as :
( ) ( ) ( )zkhnhkB znknk zφρ=ψ ρρ ,,
Cylindrical Wave Functions
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 44
Bessel functions of 1st kind
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 45
Bessel functions of 2nd kind
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 46
Bessel functions
The are nonsingular at ρ=0. Therefore, if a field is finite at
ρ=0, must be and the wave functions are
( )ρρkJ n
( )ρρkBn ( )ρρkJ n
( ) zjkjnnknk z
zeekJ φ
ρρ=ψρ ,,
The are the only solutions which vanish for
large ρ. They represent outward-traveling waves if kρ is
real. Thus must be if there are
no sources at ρ→∞. The wave functions are
( )( )ρρkH n2
( )ρρkBn( )( )ρρkH n2
( )( ) zjkjnnknk z
zeekH φ
ρρ=ψρ
2,,
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 47
( )( )
( )( )( )( ) ρ
ρ
−ρ
ρ
ρ
ρ
ρ
ρ
ρ
ρ
jkn
jkn
n
n
ekH
ekH
kρsinkN
kρcoskJ
toanalogous
toanalogous
toanalogous
toanalogous
2
1
Bessel functions
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 48
The and functions represent cylindricalstanding waves for real k as do the sinusoidal functions. The
and functions represent traveling waves for
real k as do the exponential functions. When k is imaginary (k = -jα)it is conventional to use the modified Bessel functions:
( )ρρkJ n ( )ρρkN n
( )( )ρρkH n1 ( )( )ρρkH n
2
( ) ( )
( ) ( ) ( )( )αρ−−π
=αρΚ
αρ−=αρ
+ 212 n
nn
nn
n
Hj
jJjI
( )( ) toanalogous
toanalogous αρ−
αρ
αρΚ
αρ
e
eI
n
n
Bessel functions
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 49
Circular Cavity Resonators
As in the case of rectangular cavities, a circular cavity resonator can beconstructed by closing a section of a circular wave guide at both ends withconducting walls.
ϕ
z
x
r
a
dThe resonator mode in an actual case depends onthe way the cavity is excited and the applicationfor which it is used. Here we consider TE011mode,which has particularly high Q.
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 50
,...,,,,...;q,,,...;n,,,md
zqcosmcosmsin
axJ mn
nTMmnq
32103213210 where ===
π
φφ
ρ
=ψ
,...,,,...;q,,,...;n,,,md
zqsinmcosmsin
axJ mn
nTEmnq
3213213210 where ===
π
φφ
ρ′
=ψ
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 51
The separation constant equation becomes
222
222
kdq
ax
kdq
ax
mn
mn
=
π
+
′
=
π
+
For the TM and TE modes,
respectively. Setting , we can solve for the resonant frequencies
µεπ= fk 2
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 52
d/a TM010 TE111 TM110 TM011 TE211 TM111
TE011
TE112 TM210 TM020
0.00.501.002.003.004.00∞
1.001.001.001.001.131.201.31
∞2.721.501.001.001.001.00
1.591.591.591.591.801.912.08
∞2.801.631.191.241.271.31
∞2.901.801.421.521.571.66
∞3.062.051.721.871.962.08
∞5.272.721.501.321.201.00
2.132.132.132.132.412.562.78
2.292.292.292.292.603.003.00
antdo
mnq
r
r
f
f
min
for the circular cavity of radius a and length d
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 53
mn
0 1 2 3 4 5
1234
2.4055.5208.65411.792
3.8327.01610.17313.324
5.1368.41711.62014.796
6.3809.76113.015
7.58811.06514.372
8.77112.339
Ordered zeros Xmn of Jn(X)
mn
0 1 2 3 4 5
1234
3.8327.01610.17313.324
1.8415.3318.53611.706
3.0546.7069.96913.170
4.2018.01511.346
5.3179.28212.682
6.41610.52013.987
Ordered zeros X`mn J`
n(X)
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 54
Cylindrical cavities are often used for microwave frequency meters. The cavity isconstructed with movable top wall to allow mechanical tuning of the resonantfrequency, and the cavity is loosely coupled to a wave guide with a small aperture.
The transverse electric fields (Eρ, Eφ) of the TEmn or TMmn circular wave guide modecan be written as
( ) ( ) [ ]zjzjt
mnmn eAeAzE β−β−+ +φρ=φρ E ,,,The propagation constant of the TEnm mode is
22
′
−κ=βa
x mnmn
While the propagation constant of the TMnm mode is2
2
−κ=β
ax mn
mn
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 55
Now in order to have Et =0 at z=0, d, we must have A+= -A-, and A+sin βnmd=0 or
βmnd =lπ, for l=0,1,2,3,…, which implies that the wave guide must be an integernumber of half-guide wavelengths long. Thus, the resonant frequency of the TEmnlmode is
And for TMnml mode is
22
2
π
+
′
εµπ=
dq
axc
f mn
rrmnq
22
2
π
+
εµπ=
dq
axc
f nm
rrmnq
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 56
Then the dominant TE mode is the TE111 mode, while the dominant TM mode is theTM110 mode. The fields of the TMnml mode can be written as
( )
( )
0
2
2
2
2
=
π
ρ′
′′
κη=
π
ρ′
ρ′κη
=
π
ρ′
ρ′β−
=
π
ρ′
′′
β=
πφ
ρ′
=
φ
ρ
φ
ρ
z
mnn
mn
mnn
mn
mnn
mn
mnn
mn
mnnz
E
dzq
mφa
xJ
xaHj
E
dzq
mφa
xJ
xmHaj
E
dzq
mφa
xJ
xmHa
H
dzq
mφa
xJ
xaH
H
dzq
ma
xJHH
sincos
sinsin
cossin
coscos
sincos
+−=εµ=η jAH 2 and
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 57
Since the time-average stored electric and magnetic energies are equal, the totalstored energy is
( )ρρ
ρ′
′
+
ρ′
′′
πηεκ=
φρρ
+
ε==
∫
∫ ∫ ∫
=ρ
πφρ
da
xJxma
axJ
xdHa
dzddEEWW
a mnn
mn
mnn
mn
d ae
4
2
2
02
22
2
2222
0
2
0 022
( )( )mnn
mnmn
xJxm
xdHa ′
′
−′
πηεκ= 2
2
2
2422
18
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 58
The power loss in the conducting walls is
( ) ( )[ ]( ) ( )[ ]
( )( ) ( )
′−
′
β+
′β
+′π=
φρρ=+=+
φ=ρ+=ρ==
∫ ∫
∫ ∫∫π
=φ =ρφρ
=
π
=φφ
2
2222
222
2
0 0
22
0
2
0
222
1122
002
22
mnnmmnmnn
s
a
d
zz
s
St
sc
xm
xa
xamda
xJHR
ddzHzH
dzadaHaHR
dsHR
P
( )( )
( ) ( )
′−
′β
+
′
β+
′
−
′
ηκ=
ω=
2
2222
2
2
2
3
112
1
4
mnmnmn
mn
smncc
x
mx
a
x
amad
xm
Rx
adaPW
Q
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 59
Where is the loss tangent of the dielectric. This is the same as the resultof Qd for the rectangular cavity.
( )
( )( )
δ=
ε ′′ε
=ω
=
′
′
−′
ηκε ′′ω=
ρρ
ρ′
′+
ρ′
ρ′′
πηκε ′′ω=
+
ε ′′ω=⋅=
∫
∫∫
=ρ
φρ
tan
*
1
18
4
221
22
2
2422
022
2
2
2222
22
dd
mnnmnmn
amn
nmn
nmnmn
vvd
PW
Q
xJx
m
x
Ha
da
xJ
ax
Jx
ma
x
dHa
dvEEdvEJP
To compute the Q due to dielectric loss, we must compute the power dissipated inthe dielectric. Thus,
δtan
Circular Cavity Resonators
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 60
Cavity wave guide mode patterns
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 61
Cavity wave guide mode patterns
Massachusetts Institute of Technology RF Cavities and Components for Accelerators USPAS 2010 62
Loop or Probe Coupling
For a probe coupler the electric flux arriving on the probe tip furnishes the current induced by a cavity mode:
I = ωε SEwhere E is the electric field from a mode averaged over probe tip and S is the antenna area. The external Q of this simple coupler terminated on a resistive load R for a mode with stored energy W is
In the same way for a loop coupler the magnetic flux going through the loop furnishes the voltage induced in the loop by a cavity mode: V= ωµ SH
2222
ESR
WQext
ωε=