NUCLEARINSTRUMENTS
Nuclear
Instruments
and
Methods
in
Physics
Research
A
341
(1994)
484488
8R
METHODS
IN
PHYSICS
RESEARCH
Section
A
NorthHolland
Design
of
a
30
GHz
Bragg
reflector
for
a
Raman
FEL
P
.
Zambon
,
P
.J
.M
vander
Slot
b,*
University
of
Twente,
Departmentof
Applied
Physics,
P
.O
Box
217,
7500
AE
Enschede,
The
Netherlands
n
Nederlands
Centrum
voor
Laser
Research
BV
P
.O
Box
2662,
7500
CR
Enschede,
The
Netherlands
A
design
of
a
Bragg
reflectorfora
Raman
FEL
is
described
It
is
shown
that
mode
conversion
occurs
whenever
the
axial
wavenumbers
ofthe
two
modes
fulfil
the
Bragg
condition
.
Wth
aconstant
ripple
ofthe
corrugation
it
is
shown
that
the
reflectedradiation
also
containshigher
order
modes,
assumng
that
theincidentradiation
consists
onlyofa
TE
zyxwvutsrqponmlkjihgfed
mode
.
Themode
purity
can
be
increased
by
increasingthelengthof
the
reflectoratthe
expense
of
a
smaller
reflection
bandwidth
.
A
more
flexible
method
is
byapplying
a
Hammng
window
to
the
corrugation
ofthe
reflector
.
Contributions
of
other
modes
to
thereflectedradiation
can
m
that
case
be
neglected
.
The
reflector
wll
be
installed
in
a
Raman
laser
to
be
able
to
compare
theamplifier
with
the
oscillator
configuration
.
Therefore
some
prelimnary
results
are
also
presented
about
the
startup
ofthe
Raman
laser
.
1
.
Introduction
The
Raman
type
FEL
situated
at
the
Universityof
Twente
has
so
far
been
operated
in
an
amplifier
con
figuration
with
a
maximum
current
I
b
of
the
electron
beam
of
about
900
A
maximum
[1,2]
.
In
order
to
compare
the
behaviour
of
the
laser
in
an
amplifier
and
oscillator
configuration
the
current
of
the
electron
beam
has
been
reduced
to
about250
A
maximum
[3
]
.
Wth
this
current
the
laser
can
be
operatedwith
the
two
configurations
.
Several
choices
for
the
mrrors
for
the
oscillator
areavailable
.
As
the
beam
line
contains
no
bends,
the
mrror
situated
at
the
upstream
side
of
theundulator
must
contain
a
hole
.
The
electron
beam
is
dumped
in
the
waveguide
wall
before
it
reachesthe
mrror
at
the
downstream
side
of
theundulator
.
Sincethe
laser
operates
at
wavelengths
in
the
mmregion,
one
option
forthe
downstream
mrror
is
a
mesh
.
Another
option
is
to
use
a
platewith
a
hole
in
it
through
withthe
electron
beam
passes
.
In
the
former
case
the
electron
beam
is
influenced
by
the
mesh,
but
for
the
radiation
field
itis
almost
a
perfect
reflecting
mrror,
whereas
in
the
latter
case
the
radiation
field
is
modified
by
the
hole
and
the
electron
beam
is
hardlyinfluenced
.
The
effects
of
this
hole
canbe
reduced
by
increasing
thearea
of
the
mrror,
keeping
the
hole
diameter
constant
.
However,one
then
needs
to
taper
the
waveguide
to
the
*
Corresponding
author
.
01689002/94/ 07
.00
C
1994

Elsevier
Science
B
.V
All
rights
reserved
SSDI
01689002(93)E08983
outer
diameter
of
the
mrror
to
avoid
impedance
ms
match
.
A
drawback
of
these
type
of
mrrors
is
that
there
is
no
mode
selectivity
and
it
is
difficultto
change
the
reflectivity
.
This
canbe
overcome
by
using
a
Bragg
type
reflectorconsisting
ofa
section
of
waveguide
with
a
corrugated
wall(see
e
.g
ref
.
[41
for
anintroduction)
.
The
electron
beam
will
be
undisturbed,
and
the
prop
erties
of
the
mrror,
such
as
reflected
power,
spectral
width
of
reflection
band
and
centre
frequency
of
re
flection
band,can
be
controlled
by
changing
parame
tersas
lengthof
the
mrror,
corrugation
height
and
period
of
the
corrugation
.
It
will
be
shown
that,
by
applying
a
spatial
filter,
the
mode
purity
of
the
re
flected
radiation
canbeimproved
.
The
space
availableinside
the
axial
guide
magnet
forcedus
to
choose
a
plate
with
a
hole
as
the
upstream
mrror
.
For
the
downstream
mrror
a
Bragg
reflector
is
chosen
because
it
gives
the
designer
a
flexibility
not
offered
by
the
other
typeof
mrrors
.
2
.
Design
of
a
Bragg
reflector
The
theory
of
Bragg
reflectors
is
well
known
[461
and
only
the
coupled
mode
equations
will
be
given
here
.
Consider
a
sectionof
waveguide
with
a
corru
gated
wall
for
which
theradius
is
given
by
r=r,+10
cos(kbz+00),
(1)
where
r
is
the
mean
radius
of
thewaveguide,
1,
is
the
amplitude
of
the
corrugation,
k
b
=27r/A
b
A
b
being
the
periodicity
of
thecorrugation,
and
0
is
an
arbi
trary
starting
phase
.
The
coupled
mode
equations
for
the
cylindrical
waveguide
Bragg
reflector
are
[5,6]
c
UNO
U
ô
U
m
Q
N3
0
a
1
00
0
80
0
60
040
0.20
000
P
Zambon,
P
.J
M
uander
Slot
/
Nucl
lnstr
k
z
beingthe
axial
wavenumber
of
the
incident
mode
.
Since
the
corrugation
is
azimuthallysymmetric,
the
Bragg
reflector
will
notcouple
modes
with
different
azimuthal
indices
[4]
.
The
indices
in
theaboveequa
tionsrefer
to
the
radial
indicesof
the
modes
.
f,+
is
the
amplitude
of
the
(incident)
waveguide
mode
propagat
ing
in
the
forward
direction
and
f,
is
the
amplitude
of
the
backward
propagating
(reflected)
wave
.
G
zyxwvutsrqponmlkjihgfedcbaZYXWVUT
G,
P
,
and
P
arethe
coupling
coefficient
between
the
for
ward
and
backward
components
of
the
incident
mode,
the
coupling
coefficient
between
two
TE
or
TM
modesand
the
crosscoupling
coefficient
between
a
TE
(H)
and
TM
(E)
mode
respectively
.
Expressions
for
these
coefficients
can
befound
inrefs
.
[5,6]
.
Solving
thesystem
of
equations
given
by
(2)
and
(3)
fromz
=
0
to
z
=
L,
L
beingthe
length
of
the
mrror,
one
finds
the
mode
amplitudes
fromwhich
the
reflection
R
and
25
30
354045
and
Meth
.
m
Phys
.
Res
.
A
341
(1994)
484488
485
transmssion
T
coefficients
for
each
mode
can
be
cal
culated
R=1f
(z=0)I
Z
/If+(z=0)I
Z
,
T=1f+(z=L)Iz/If+(z=
(»
Iz
.
The
phase
shift
of
the
reflected
(¢,)
and
transmtted
(0,)
components
of
awaveguide
mode
can
also
be
obtained
fromtheamplitudes
f,
+and
f,
0,
=
tan'(f
n,(zL)/fc(Z
L)),
(hT=tan'(f,»(z0)/fe(z0)),
(8)
where
the
subscripts
im
and
re
stand
for
imaginary
and
real
part
of
the
complex
amplitude
.
In
the
Raman
laser
the
FEL
interaction
basically
takes
place
with
the
TE
mode,
though
cyclotron
instabilities
can
result
in
interaction
with
other,
higher
order,
modes
[2]
.
The
Bragg
reflector
must
therefore
reflect
the
TE
mode
and
preferably
none
of
the
other
modes
The
central
frequency,
chosen
to
be30
GHz,
is
given
by
the
Bragg
condition,
k
Z
=k
h
/2
.
The
periodof
the
corrugation
for
reflection
of
the
TE
mode
at
30
GHz
thus
becomes
.t
i
,
=
5
.37
mm
The
system
of
equations
(2)
and
(3)
canbe
solved
for
any
number
of
modes,
but
for
practical
reasons
i
.e
.
for
reasonable
computing
time)
the
number
of
modes
simultaneously
present
is
limted
to
two
in
the
(mod
ified)
code
[5]
used
.
The
incident
mode
atz
=
0
is
assumed
to
beapure
TE
S
mode
whereas
theother
mode
is
chosen
to
be
the
TM
t
mode
.
The
reflector
is
matched
at
z
=
L,
i
.e
.
nowaves
arereflected
at
the
end
of
themrror
.
The
power
reflectioncoefficient
is
shown
in
Fig
.
l
a
forthecase
of
L
=
120
mm
and
1,
=
0
.5
mm
.
The
transmssion
coefficient
is
shown
in
c
1
00
m
UO
U
(D
0
80
ô
0
60
NNy
040
C
c0
020
3
IL
0
00
Frequency
(GHz)
Frequency
(GHz)
(5)(6)
25
30
354045
Fig
1
.
Power
reflection
coefficient(a)
and
transmssion
coefficient
(b)
forthe
TE
I
mode
(solid
line)
and
TM
I
mode
(dashed
line)
for
the
Bragg
reflector
versus
frequency
.
The
incident
mode
is
assumed
to
be
a
pure
TE
mode
.
The
parameters
of
the
Bragg
reflector
are
d
n
=
5
.37
mm
1,
=
0
.5
mm
and
L
=
120
mm
IX
OPTICAL
TECHNOLOGY
'f 
_
id,,f
+
iG
f,
z

iY_H,1,f,,P
 
i
Y_
G,fti,
P
,
(2)
P
_dfr
_
JA
f=
1G
f
+
iY_H
P
ft'
.
P
c3z
P

J
Y_
G,PfH,P
>
(3)
P
where
=k
z
kt,/2,
(4)
486
P
Zambon,
P
.J
.M
cander
Slot
I
Nucl
.
Instr
.
and
Meth
.
i
n
Phys
Res
.
A
341
(1994)
484488
Fig
.
l
b
.
It
can
clearly
be
seen
that
the
incident
TE
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPON
mode
is
primarilyreflected
asa
TE,
1
mode
at
30
GHz
(FWHM
=
1
.7
GHz)
and
as
a
TM
t,
mode
around
33
.4
GHz
(FWHM
=
2
.4
GHz)
.
Note
that
the
Bragg
reflectoractsas
a
mode
converter
atthis
frequency
which
is
given
by
the
Bragg
condition
kT
E
+kZ
M
=k
B
.
(10)
At
the
designfrequency
of
30
GHz
not
all
the
power
is
reflected
in
the
TE
mode
because
of
the
side
lobesof
the
TM
reflection
.
Almost
all
the
trans
mtted
power
is
in
the
TE
mode,
while
only
a
mnor
fraction
is
transmtted
in
the
TM,
mode
(see
Fig
.
1b)
#'
.
The
side
lobes
may
be
undesired
in
high
gain
FELs
.
One
way
to
decrease
the
effects
of
side
lobes
is
to
make
the
reflection
band
around
the
centre
fre
quency
smaller,
by
increasing
the
length
of
the
mrror,
so
that
overlap
between
reflections
will
be
less
.
A
different
way
to
decrease
or
even
remove
the
effects
of
side
lobes
and
to
obtainhigh
mode
purity
is
to
apply
a
Hammng
window
to
thecorrugated
partof
the
mrror
.
This
is
particularly
interestingfor
over
moded
waveguides
where
higher
order
modes
are
not
damped
.
This
typeof
waveguide
is
often
used
in
FELs
because
propagation
of
the
electron
beam
requires
waveguide
dimensions
corresponding
to
overmoded
operation
for
the
radiation
generated
.
A
Hammng
window
can
be
applied
to
the
Bragg
reflector
by
modi
fying
the
corrugation
height
1
according
to
Trz
1=100
.540
.46*
cos
2
(
~~,
(ll)
where
L
is
the
length
of
thecorrugated
section
which
starts
atz
=
0
.
The
Raman
FEL
for
which
this
mrror
is
designedhas
anelectron
beam
duration
of
100
ns
.
The
radiation
field
can
thereforeonly
haveabout
6
round
trips
during
which
the
electron
beam
is
present
.
As
thesystem
starts
from
noise,
and
as
we
want
to
investigate
the
difference
between
an
amplifier
and
oscillator
con
figuration
it
was
decided
to
use
a
power
reflectivity
around
the
40
forthe
TE
mode
at
the
design
frequency
of
30
GHz
.
The
single
passgain
will
be
changed
in
the
futureby
varying
the
lengthof
theundulator
.
Again
thesystem
of
coupled
equations
is
solved
with
the
Hammng
window
applied
to
the
reflector
with
a
corrugation
amplitude
of
0
.2
mm
The
power
reflectioncoefficient
is
shown
in
Fig
.
2
.
The
side
lobes
havedisappeared
and
the
reflection
bands
forthe
twomodes
are
completely
separated
.
Even
for
the
0
.5
mm
ripplereflector
one
finds
that
the
two
modes
are
still
t
This
is
a
general
property
of
the
reflector,
i
.e
.
the
mode
ofthe
transmtted
radiation
is
basically
theincident
mode
.
U
m
0
U
ô
U
m
Q
m
3
0
1
000750
.50
0250
.00
25
30
35
40
Fig
.
2
Power
reflection
coefficient
for
the
TE
and
TM
mode
The
parameters
ofthe
Bragg
reflector
are
Ab
=
5
.37
mm
t
o
=
0
.2
mm
and
L=
270
mm
(solid
line)
and
1
=
0
.17
mm
(dashed
line)
.
The
crosses
indicate
measured
valuesofthe
reflection
coefficient
.
completelyseparated
if
the
Hammng
window
is
ap
plied
.
3
.
Measurement
of
reflection
properties
of
the
Bragg
reflector
A
Bragg
reflector
with
A,,
=
5
.37
mm,
1,
=
0
.2
mm
and
L=
270
mm
has
been
constructed
with
the
Ham
mng
window
(7)
applied
to
the
ripple
.
The
reflection
properties
are
measured
in
order
to
validate
the
calculations
made
.
The
setup
is
schematically
shown
in
Fig
.
3
.
A
HewlettPackard
model
HP8690B
K
.band
sweep
generator(26
.540
.0
GHz)
and
a
model
HP8755C
am
plitude
analyser
areused
to
generate
the
input
power
and
analyse
the
forward
and
backward
propagating
power
which
are
measured
usingonedirectionalcou
plers
.
The
Raman
FEL
utilises
a
cylindrical
waveguide
.
Onedirectional
CouplerForward
Power
Amplitude
Analyser
frequency(GHz)
Rectangular
to
CircularTransition
Onedirectional
Coupler
BackwardPower
Cylindrical
TaperOutcouple
Horn
Fig
.
3
.
Schematic
overview
ofthe
setup
used
for
measuring
the
power
reflection
of
the
Bragg
reflector
P
Zambon,
P
.J
M
van
der
Slot/
Nucl
.
Instr
and
Meth
.i
n
Phys
.
Res
A
341
(1994)
484488
Therefore
a
rectangular
to
cylindrical
waveguide
tran
sition
is
needed
togetherwith
a
cylindrical
taper
to
switch
from
standard
K
a
banddimensions
to
thecus
tom
size
of
the
Bragg
reflector
.
The
rectangular
to
cylindricaltransition
converts
the
rectangular
TE,
waveguide
mode
to
a
cylindrical
TE,
mode
.
Higher
order
modes
present
in
the
cylindrical
section
of
the
measurements
willin
principle
notcouple
to
the
fun
damental
TE,
o
rectangular
waveguide
mode
asthey
are
reflected
at
the
transition
.
However
the
Bragg
reflector
is
reversible,
i
.e
.
if
ata
frequency
fo
an
incomng
TE,
mode
is
reflectedinto
a
TM
t
mode
then
at
the
same
frequency
an
incomng
TM
t
mode
will
be
reflected
in
a
TE
zyxwvutsrqponmlkjihgfedcbaZYXWVUTSRQPONMLKJIHGFEDCB
mode
.
Thus
higherorder
modes
canbe
measured
with
this
setup,
though
the
reflected
power
will
depend
on
the
double
mode
conversion
that
has
takenplace
.
One
side
of
the
mrror
is
connected
to
the
cylindrical
taper
while
at
the
other
side
an
outcouple
horn
(used
in
the
Raman
FEL)
is
mounted,
i
.e
.
the
reflector
is
matched
at
thatside
and
there
are
no
reflections
.
Spurious
reflectionsinside
the
measurement
system
limted
the
dynamc
range
to
about
20
dB
.
The
results
are
also
shown
in
Fig
.
2
.
The
centre
frequencies
of
the
two
reflection
bands
clearly
coincidewith
the
calcu
lated
ones
.
The
total
reflectivity
measured
is
somewhat
less
than
calculated
.
The
amplitude
ripple
used
in
the
calculation
was
0
.2
mm
The
peak
reflectivityat
30
GHz
is
reduced
to
25
for
a
corrugation
heightof
0
.15
mm
whereas
it
is
30
for
t
o
=
0
.17
mm
The
calculated
reflection
coefficientfor
the
latter
value
is
also
shown
in
Fig
.
2
.
For
the
TE,
mode
goodagreement
is
found
between
the
measured
values
and
the
calculationfor
t
o
=
0
.17
mm
The
double
mode
conversion,
in
the
experiment,
for
the
TM,
mode
should
reduce
the
measured
values
to
87
of
thetheoretical
ones
.
One
can
thus
state
thatwithin
themechanical
limts
of
making
the
corrugation,
good
agreement
is
found
be
tween
thecalculations
and
the
measurements
for
the
TE,
and
the
TM
modes
.
4
.
Startupof
the
Raman
FEL
In
this
section
some
prelimnary
results
will
be
presentedobtainedwith
a
numericalcode
.
The
code
used
is
basically
the
one
described
in
ref
.
[7]
.
The
radius
of
the
electron
beam
is
such
that
the
assumption
ofperfect
orbits
and
that
of
neglecting
the
radial
dependence
of
the
undulator
field
is
not
valid
.
Themodel
has
been
modified
to
include
the
full
three
dimensional
electron
orbits
.
The
current
version
2
2
The
model
wll
be
described
in
more
details
in
a
future
publication
.
m30
a
0
0m30
a
0
0
Z
(m
Fig
.
4
.
The
logarithmcofthe
power
(solid
line)
and
the
radiation
phase(dashed
line)
as
afunctionofthedistance
along
the
undulator
.
The
initial
power
is
assumed
tobezero
while
the
electronsare
initialised
with
a
uniform
phase
distri
bution
with
a
small
random
disturbance
(TDAcode)
.
The
main
simulation
parameters
areI
=
200
A,
B
=
0
.18
T,
Au
=
0
.03
m
B
z
=1
.0
T,
E
=
10S,R
mrad
and
y=2assumesan
idealhelical
undulator
field
.
For
now
par
ticular
attention
has
been
given
to
the
startup
of
the
laser
.
Therefore
the
initial
radiation
amplitude
is
as
sumed
to
be
zero
in
thecalculations
and
the
laser
starts
from
noise
in
theelectron
beam
Two
methods
ofelectron
beam
initialisation
havebeenused
.
One
is
the
initialisation
used
by
the
FRED
codeand
the
other
is
used
in
the
TDA
code
[8]
.
The
TDAinitialisation
has
been
modified
tostart
withzero
radiation
field
and
a
uniformphase
distribution
modified
bya
small
random
number
[9]
.
The
code
has
beenrun
for
theparameters
of
the
Twente
Raman
FEL
[2]
with
an
undulator
field
of
0
.18
T,
a
guide
field
of
1
.0
T,
a
current
of
200
A
and
a
normalisedemttance
of105
rr
mrad
.
Growth
is
found
for
a
radiation
frequency
of
30
GHz
.
The
growth
and
the
phase
of
the
radiation
field
is
given
in
Figs
.
4
487
m
L
a

Z
(m
Fig
.
5
.
As
Fig
.
4
except
that
theelectrons
are
initialisedin
the
sameway
as
in
the
FREDcode
.
IX
.
OPTICAL
TECHNOLOGY
488
5
.
Conclusions
P
Zambon,
P
.J
.
AI
van
der
Slot
/Nucl
Instr
and
Meth
.
t
o
Phys
.
Res
.
A
341
(1994)
484488
and5
forthe
TDA
and
FREDinitialisation
respec
tively
.
The
lethargy
i.e
.
the
distance
before
exponen
tial
growth
starts)
is
aboutthe
same
for
both
initialisa
tions
.
The
FREDinitialisation
gives
however
a
lower
field
amplitude
at
the
beginning
of
the
exponential
growth
region
resulting
in
a
factor
5
loweroutput
at
the
end
of
theundulator
.
The
TDAinitialisation
resultsin
saturation
atalevelof
about
12
MW
whereas
this
is
notthe
case
for
the
FREDinitialisation
.
The
phase
of
the
radiation
field
changes
more
rapidly
in
the
lethargyregion
for
the
FREDinitialisation
than
for
the
TDAinitialisation
whereas
during
exponential
growth
thefrequency
shift
is
slightlyless
.
By
varying
the
lengthof
theundulator
it
becomes
possible
to
investigate
thebehaviour
ofanamplifier
and
oscillator
configuration
for
differentsingle
passgains
.
A
Bragg
reflector
has
been
designed
with
a
power
reflection
coefficient
of
=
0
.4
at
the
design
frequency
of
30
GHz
for
an
incomng
and
reflected
wave
in
the
TE
tt
mode
.
By
applying
a
Hammng
window
to
the
corrugation
itis
possible
toobtain
high
mode
purity
and
still
maintain
arelative
large
FWHM
of
the
reflec
tion
peak
.
Itis
found
that
the
reflectivity
is
a
sensitive
functionof
the
ripple
amplitude
for
the
parameter
region
investigated
.
A
Bragg
reflector
has
been
constructedwith
X1
6
=
5.37
mm,
1,
=
0
.2
mm
and
L
=
270
mm
Allowing
for
mechanical
tolerances
in
making
the
corrugation,
good
agreement
is
found
between
the
measured
and
calculated
reflectivity
.
Prelimnary
calculations
show
that
the
laser
may
just
orjust
not
saturate
in
a
single
pass
dependingon
the
typeof
initialisation
used
for
the
electron
beam
in
the
simulation
.
Varying
the
interactionlength
makes
it
possible
to
investigate
the
different
behaviour
of
an
oscillator
and
amplifier
configuration
.
Acknowledgements
The
authors
would
like
to
thank
P
.J
.S
.
Teunisse
from
Hollandse
Signaalapparaten
B
.V
.,
The
Nether
lands
and
A
F
.M
Bouman
for
their
assistance
with
the
rf
diagnostics
.
References
[1]
P
.J
.M
vander
Slot,
Ph
.D
.
Thesis,Universityof
Twente,
The
Netherlands
(1992)
.
[2]
P
.J
.M
vander
Slot
and
W
J
Wtteman,
Nucl
Instr
andMeth
A
313
(1993)
140
.
[3]
P
Zambon,
W
J
.
Wtteman
and
P
.J
.M
vander
Slot,
these
Proceedings
(15th
Int
.
Free
Electron
Laser
Conf
.,
The
Hague,
The
Netherlands,1993)
Nucl
.
Instr
and
Meth
.
A
341
(1994)
88
.
[4]
V
.L
.
Bratman,
G
.G
.
Denisov,
N
.S
.
Gnzburg
and
M
I
.
Petelin
.
IEEE
J
.
Quantum
Electron
.
QE19
(1983)
282
.
[5]
J.C
.
Cheng,
B
.S
.
thesis,
MT,
USA
(1991)
.
[6]
G
.G
.
Denisov
and
M
G
.
Reznikov
.
Izv
.
Vyssh
.
Uchebn
.
Zave
d
Radiofiz
.
25
(5)
(1982)
562,translated
by
AIP
.
[7]
J
.S
.
Wurtele,
R
Chu
and
J
.
Fauns,
Phys
Fluids
B
2(1990)1626
.
[8]
Both
typesof
initialisation
are
present
in
TDA
Version
0
.3

T.M
Tran
and
J
.S
.
Wurtele,
Comp
.
Phys
.
Commun
.
54
(1989)
263
.
[9]
C
Penman
and
B
.WMcNed,Opt
.
Commun
.
90(1992)82