Transcript Muralidhar Yeddulla Design of Microwave Undulator Cavity

Design of Microwave
Undulator Cavity
Muralidhar Yeddulla
Physics of Undulator
• Highly relativistic electron beam passing
through a pump field (wiggler field) produces
synchrotron radiation
• Combination of radiation and wiggler field
produce a beat wave which tends to bunch
electrons in the axial direction
• The bunched electrons radiate coherently
producing quasi-monochromatic synchrotron
radiation
Why microwave undulator
• Limitations of permanent magnet undulators
– Polarization cannot be controlled
– Undulator period cannot be changed
• Advantages of a microwave undulator
–
–
–
–
No magnets to be damaged by radiation
Small undulator periods and larger apertures possible
Beams with larger radius and emittance can be used
Helical undulators are relatively easy with microwaves
• Drawbacks of a microwave undulator
– High power microwave sources with precise and stable
amplitude and phase are expensive
– Handling of tens of GW of microwave power is challenging
– Design of waveguide/cavity structure can be complicated
Applications of polarization control
• Exciting scientific opportunity in areas
where scattered or absorbed x-ray signal
from sample depends on polarization state
• For example, measurement of very small
magnetic moment changes in magnetic
devices requires fast modulation and lockin techniques to suppress systemic errors
caused by slow drifts which is not possible
with pump-probe techniques
Planar Microwave Undulator
Vector plot of deflecting field in mid-plane of the waveguide
Helical Microwave Undulator
Right
Left
Vector plot of deflecting field in mid-plane of the waveguide
Microwave undulator realization
• High power handling capabilities of the
undulator waveguide have to be
maximized and losses in the waveguide
minimized
• The operating microwave mode should
have maximum field strength in the path of
electron beam and minimum tangential RF
magnetic field near waveguide wall
Structure of microwave undulator
d x eBu
2 z

cos
dt
mo
u
d x eBu
2 z

cos
dt
mo
u
d x
e

( Ex  vz By )
dt
mo c
Bu 
Eo
c
Eqn. of motion for magnetic undulator
Eqn. of motion for microwave undulator
Need for microwave cavity
Magnetic field in conventional static undulator:
B y  Bu cos
2 z
u
• To achieve comparable field strength By, tens of GW
of RF power is needed.
• RF energy can be stored in a cavity by pumping in
RF power to achieve the required field strength By
• The required field strength can be sustained by only
compensating for RF losses in the cavity
Theory of microwave undulator
Electron bunch
• Electrons can interact with both forward and
backward flowing RF waves
• Cavity dimensions are usually large to keep
losses low and to reduce sensitivity to
dimensions
Calculated flux curves for BL13
Elliptical Polarized Undulator
3.4 x 1015
# of periods = 65, λu = 5.69cm, K=1.07, Beam current = 500 mA
Requirements of SPEAR ring RF
undulator to be designed
• Electron beam energy: 3 GeV, beam
current: 500 mA, aperture: 2 x 2 mm
• Radiation energy: 700-900 eV (circular
polarization)
• Fast polarization switch-ability
• Average photon flux and brightness to be
within a factor of 10 compared to BL13
static magnetic field Elliptical Polarization
Undulator
Choice of waveguide and modes
• RF field should be “rotate-able” to control
radiation polarization
• We consider only circular symmetric waveguide
• Operating RF mode should have strongest field
on waveguide axis along path of electron beam
• Very important to keep wall losses extremely
low
• Wall losses determine the cost and feasibility of
microwave power source
Waveguide modes considered
Cylindrical waveguide
TE11 mode
TE12 mode
Corrugated waveguide
HE11 mode
Circular waveguide mode TE11
• Fundamental mode, easily
excitable
• Has very strong RF field
on the axis where the
electron bunch travels
• Not the least lossy mode
Electric field over waveguide cross-section
Power loss in a TE11 circular
waveguide undulator
Power Flow, GW
9
8
7
6
5
4
5.5 6.0 6.5 7.0 7.5 8.0
Power Loss, MW
19.2
19.1
19.0
18.9
18.8
18.7
18.6
5.5 6.0 6.5 7.0 7.5 8.0
Waveguide radius, cm
• Flux = 1/5th of Static
undulator flux (3.4 x 1015
[ph/s/0.1 % BW)
• K = 0.71
• Circular polarization
• Photon energy = 700 eV
• L = 3.7 m
• λu = 6.1 cm
• f = 2.6 GHz
Circular waveguide mode TE12
• Has same field structure
at the axis as TE11 mode
• Needs much larger
waveguide radius and
power
• Attenuation is much lower
than TE11 mode
Electric field over waveguide cross-section
Power loss in a TE12 circular
waveguide undulator
Power Flow, GW
350
300
250
200
150
100
40
50
60
Power Loss, MW
6.3
6.2
6.1
6.0
5.9
5.8
40
50
60
70
70
Radius, cm
80
80
• Flux = 1/5th of Static
undulator flux (3.4 1015
[ph/s/0.1 % BW)
• K = 0.68
• TE12 mode, Circular
polarization,
• Photon energy = 700 eV
• L = 3.7 m
• λu = 6.35 cm
• f = 2.38 GHz
Corrugated waveguide, hybrid HE11
mode
2a
2b
L
• Combined TE and TM modes lead to hybrid modes
• Under “balanced hybrid” conditions, the field transforms in to low loss linearly
polarized wave
• The field is strongly linearly polarized on the axis which is highly desirable for
undulator operation
• The field is extremely low near the waveguide walls translating to very low RF losses
β L = 0.05 rad, Q = 36945
β L = 0.3 rad, Q = 42537
β L = 0. 5 rad, Q = 74322
β L = 2 rad, Q=13574
β L = 1 rad, Q = 1.05 x 106
β L = 1.65 rad, Q=5.5 x 106
Analysis of corrugated waveguide
(neglecting space harmonics)
Fz  A J m kr r e
jm
r<a
; Az  AZ z J m kr r e
kr2
kr2
Ez   j
Az ; H z   j
Fz
k 
k 
1 Fz
 z Az
Er  

 r  k  r
• Both TEz and TMz modes
are present
• Balanced Hybrid mode is
possible only when wave
guide impedance Zz ≈ Free
space wave impedance
jm
a<r<b

H m( 2) krH m(1) kb  H m(1) krH m( 2) ka jm
Az  jA
J m kr a  ( 2)
e
(1)
(1)
( 2)
k
H m kaH m kb  H m kaH m kb
k
1 Az
1 Az
Ez   j
Az ; H r 
; H  
r 
 r

• Only TMz mode present
• Balanced Hybrid condition possible
when (b – a) ≈ λ/4
Boundary conditions for corrugated
waveguide
• For sufficiently small slot width, no TEz
mode present inside corrugation
• Then, at r = a, EΦ = 0
• Admittance HΦ /Ez is continuous at r = a
• Equating admittance for the two set of
equations at r = a gives the characteristic
dispersion equation for the corrugated
waveguide
Power loss dependence on
corrugation depth
Total Power, GW
225
220
215
210
205
200
0.86 0.88 0.90 0.92 0.94 0.96
Power Loss, MW
10
9
8
7
6
5
4
0.86 0.88 0.90 0.92 0.94 0.96
a/b
• Flux = Static undulator flux
(3.4 1015 [ph/s/0.1 % BW)
• K=1.07
• Photon energy = 700eV
• L = 3.7 m
• b = 23 cm
• f = 4.05 GHz
• λu = 5.92 cm
Power loss in a HE11 corrugated
waveguide undulator
Total Power, GW
500
400
K = 1.07
300
200
100
0
15 20 25 30
Power Loss, MW
10
8 K = 1.07
6
4
2
0
15 20 25 30
• Flux = Static undulator flux
a/b = 0.9
• K = 1.07
• f = 4.05 GHz
K = 0.73
• λu = 3.71 cm
35 40 45 50
a/b = 0.9
• Flux =1/5th Static undulator
flux
• K = 0.68
K = 0.73
35 40 45 50
Radius b, cm
• f = 2.55 GHz
• λu = 5.92 cm
Superiority of HE11 - mode
TE11
TE12 HE11
0.71
0.68 0.68
5.8
180
79
RF power loss
5.1
(MW/m)
RF frequency (GHz) 2.64
1.6
0.326
Cavity Radius (cm)
57.7 38
Undulator
parameter K
Power flow (GW)
6.5
2.38 2.37
Design of cavity ends
• For 1/5th Static undulator flux, power loss in short circuited
end plates is 10.5 MW!
• Loss in the ends is 8-9 times greater than in the whole length
of 3.7m of the corrugated waveguide!
• Better low loss ends needed
Varying depth corrugations for
cavity ends
• Cavity ends to be designed for power reflection
• Corrugation depths and width vary with distance
• The RF field pattern likely to change from low loss HE11
mode pattern near the ends and has to be taken into account in
the design
Mode Matching for corrugated
waveguide (includes space harmonics)
E I   ( FmI  BmI )emI
m
FI
2b
2a
I
e
H I   ( FmI  BmI ) mI
Z gm
m
FII
E II   ( FnII  BnII )enII
BII
n
BI
II
e
H II   ( FnII  BnII ) nII
Z gn
n
At the interface, E I  E II .
 F
m
I
 B I  emI   F II  B II  enII , a  r  0
n
 H I  wall
,b  r  a
 BI 
 FI 
 II   S  II 
B 
F 
 


Issues with varying depth
corrugation ends
• Length of the ends is at least twice the
diameter of the waveguide (D=76 cm) for
mode conversion to HE11 mode
• The end length is unacceptably long for
the limited length available in the SPEAR
ring for the undulator
• The field structure in the ends varies with
length which will lead to a wide spread in
the radiation spectrum
Ideas to reduce end losses
z=L1
• RF magnetic fields near the end walls should be minimized
• Corrugating end walls to cut off propagating modes reaching the end
• Phase cancellation such that maximum RF power reflects from regions away
from the end
• Dielectric loading
Ridge ends along electric field lines
Smooth Rectangular cavity
Rectangular cavity with Ridge ends
Q = 74,500
Q = 45,500
• Most losses occur on the surface of the ridge due to concentration of RF fields
• Reducing the ridge width does not improve Q
Conclusions and future work
• Corrugated waveguide is an attractive option for
a microwave undulator due to its low losses
• However, simple short circuit ends for the cavity
leads to close to an order greater conductor loss
than in the 3.7m length waveguide
• The length of a tapered corrugated end is too
long to be useful
• Considering various end structures to minimize
loss in ends while not affecting the balanced
hybrid HE11 field structure