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Radio-‐frequency and mm-‐wave sources
• accelera&ng cavity needs to be excited by radiofrequency ](RF) or mm waves.
• RF are typically produced in – klystrons – magnetron
• high frequency (mm) waves are produced in klystrons and other type of “back-‐ wave oscillators”
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 1
linac
RF source
waveguides Courtesy C. Ng (SLAC)
Principle of an RF source
• transform a dc electron beam into a train of bunches
• use this train to excite a radia&ve process • emiUed radia&on is coherently enhanced at the bunching frequency (see HW3)
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 2
Bunch form
factor
Appl. Phys. LeU
. 98, 261501 (2011)
RF source – historical context • method to produce e-‐ beam with “varying density” conceived by D. Rozhansky from Leningrad Polytechnic Ins&tute (1935)
• UK patent by O. Heil (1935) • theory of velocity modula&on and beam bunching by A. Arsenjewa-‐Heil, and O. Heil, Z. f. Physik 95 (1935)
• Varian brothers build a klystron (1937)
• Hansen (SLAC) invents the first cavity
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 3
A simple klystron configura@on
• (see HW3)
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 4
Principle of opera@on
• c
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 5
Example of klystrons
• .
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 6
Simple theory
• 1st stage: produc&on of a dc electron beam – the beam kine&c energy is given by the applied DC voltage:
• 2nd stage is to apply a &me-‐dependent voltage on the bunch
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 7
1
2mu2
0 = eV0
Simple theory: electron emission
– the change in kine&c energy is therefore
• the final velocity is
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 8
1
2mu2 � 1
2mu2
0 = eV1 sin!t
u(t) = u0
r1 +
eV1
V0sin!t
@me of arrival at “output gap”
• assume • consider an electron that get an “instantaneous” velocity modula&on at
• if is the separa&on between the two “gaps” the the &me of arrival at the 2nd gap is
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 9
t1
V1/V0 ⌧ 1
l
t2 = t1 +l
u' t1 +
l
u0� lV1
2u0V0sin!t1
Final current
• introducing a phase • the laUer equa&on rewrites • the charge conserva&on requires
• but
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 10
' ⌘ !t
!t2 = !t1 + ✓0 �X sin!t1
I1dt1 = I2dt2
dt2dt1
= 1 +X cos!t1
Final current
• so that the final current can be wriUen
• current can in principle be infinite • we can relate this with our 1st order beam dynamics approach (introducing the R56 of a drie space… )
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 11
I2 =
I11�X cos!t1
final current • the final current is limited by field nonlineari&es and space charge effects
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 12
final current • In prac&ce the final current is wriUen as the Fourier series:
• where
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 13
I2 = I1 +1X
1
an cosn(!t2 � ✓0)
an =
1
⇡
Z ✓0+⇡
✓0�⇡cos[n(!t2 � ✓0)]d(!t2)
=
I1⇡
Z ⇡
�⇡cos[n(!t1 �X sin(!t1)]d(!t1)
= 2I1Jn(nX)
efficiency (at fundamental f)
• maximum current is • for so that output power at the fundamental frequency (n=1) is
• ~60% efficiency conver&ng DC power into the fundamental harmonic
• addi&onal contribu&on comes from the output coupling (power extrac&on from the cavity)
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 14
max[I] = 2⇥ 0.58X = 1.84
Pout
=1.16I1p
2
V0p2= 0.58P
in
other source magnetron • cheaper and more compact than klystrons but harder to stabilize
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 15
anode block
coaxial coupler (power extrac&on)
cathode resonant cavi&es
interac&on space
magnetron opera@on
• velocity and density bunching happens axially
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 16
hUp://www.radartutorial.eu/08.transmiUers/Magnetron.en.html
backward wave oscillators
• slow wave structure used to bunch the beam
• an evanescent back-‐propaga&ng wave (nega&ve group velocity interacts with the dc beam to produce a bunch train)
PHYS 790-‐D Special topics in Beam Physics, Fall 2014 17
