July 11, 2001
CLNS 01/1758
_____________________________________________________________________________________


POLARIZED e , e PRODUCTION BASED ON CONVERSION OF
GAMMAS, OBTAINED FROM HELICAL UNDULATOR
A TALK AT SNOWMASS2001
Alexander A. Mikhailichenko
Cornell University, Ithaca, NY 14850
1. Preface
2. The concept of conversion system for polarized e  , e  production
Interaction of circularly polarized gammas with media
3. Undulator radiation
Number and polarization of quants for different harmonics
4. Efficiency
Analytical calculations
Numerical codes for gamma-positron production
5. Undulator design
Codes for the field calculations
Tested undulators with period 6 and 10 mm
6. Technical details of collection system
Lithium and solenoidal short focusing lenses
Energy deposition in a target
Light target
7. Perturbations
Emittance and spin perturbation in an undulator
Spin perturbation at interaction point
Resistive instability in undulator chamber
Heating
Irradiation of walls
8. Undulator length reduction
Few targets in series
Form-factor for Ti target
9. Proposal for test at SLAC accelerator
Undulator Converter Test Facility –UCTF
SLAC B-Factory
10. Conclusion
____________________________________________________________________________________
See:


1)A.A. Mikhailichenko, Polarized positron sources, International Workshop on e , e Sources and PreAccelerators for LC, SOURCES 94, Schwerin, Germany, September 29 - October 4, 1994.
2)A.A. Mikhailichenko, Use of undulators at High Energy to Produce Polarized Positrons and Electrons,
Workshop on New Kinds of Positron Sources for Linear Colliders, SLAC, March 4-7, 1997, SLAC-R-502,
pp.229-284.
3)DESY publications: J. Rossbach, K. Floetmann, DESY 93-161; DESY M-91-11; R. Glanz, DESY 97-201.
1
1. PREFACE
Spin of particles remains an enigma and this will be so for incoming years. Particles with certain
spin states before interaction represent clean initial conditions. This will help much in solving this
enigma. Both highly polarized colliding particles must be a key instrument in all future
investigations at high-energy colliders.
This is especially true beyond 100 GeV where electromagnetic and weak interactions become
comparable.
High-energy states are polarized ones, so each particle from the first bunch needs to be collided
only with appropriately polarized partner from incoming bunch to build up an intermediate state of
interest. So making both polarized beams one can increase productivity at least two times.
FIGURE 1: Polarized colliding bunches can build up intermediate states with any particular spin.
High-energy colliders to be effective require high positron/electron population, so the source of
polarized positrons must deliver them in sufficient amount and quality.
Target itself has a tendency to overheating during positron production with ordinary schemes,
based on developing of shower in a thick heavy target.
Examples of recent day’s description of the collisions with polarized particles one can find in [1].
[1] A.A. Likhoded, M.V. Shevlagin, O.P. Yushchenko, Mass bounds on the extra neutral vector bosons at future
e  , e  colliders with polarized beams, International Journal of Modern Physics A, Vol. 8, No. 28 (1993), pp.
5063-5077.
 
A.A. Likhoded, T. Tan, G. Valencia, Study of Anomalous coupling at a 500 GeV e e linear Collider with Polarized
Beams, Phys. Rew D, vol 53, N 9, May 1996, pp. 4811-4821.
Cross section of colliding polarized particles can be written as the following
4 2
 ( Pz , Pz ) 
 ( A1 S1  A2 S 2 ) ,
3s
where S1  1  Pz Pz , S2  Pz  Pz . For s  500GeV
e  e        A1=1.13, A2=0.08
e  e   hadrons  A1=6.45, A2=3.03
For Pz  0.8 , Pz  0.7 , S1  1.56 , S2  1.5 what ~ doubles cross section.
Importance of polarization for seeking new bosons beyond the Standard Model was stressed there.
Predictions made that at the energy range s  500GeV , the polarized beams give the luminosity
gain by ~ 5 times or with unpolarized beams the total energy needs to be 2-3 times higher.
Importance of polarization was recognized at BINP-VLEPP from the very beginning of
Linear Collider activity, in earlier 1979. In mid 1979 the self-consistent scheme was
proposed.
2
2. THE CONCEPT OF CONVERSION SYSTEM FOR POLARIZED
e  PRODUCTION
Idea was developed in [2] to convert the circularly polarized gammas of
appropriate energy into longitudinally polarized positrons/electrons in a thin
target.
[2] V.E. Balakin, A.A. Mikhailichenko, Conversion system for obtaining highly polarized electrons and positrons,
Preprint INP 79-85, Novosibirsk 1979.
V.E. Balakin, A.A. Mikhailichenko, VLEPP: the conversion system, Proc. of the 12 Int. Conference on high
energy Accelerators, Batavia, 1983, p. 127.
Circularly polarized  -source
Target
Collection
Selection
FIGURE 2: The basis of conversion system for polarized particle production.
So the method realizing this idea [2] includes (Fig. 2):
Irradiation of thin (in terms of radiation length) target with circularly polarized gammas having
sufficient energy;
Collection positrons created in a target near the top of theirs energy.
Due to specific properties of the photon interaction with the matter, the positron/electron at
the high top energy spectra have longitudinal polarization.
[3] V.N. Baier, V.N. Katkov, V.S. Fadin, Radiation of the relativistic electrons, Moscow, Atomizdat, 1973.
The sources of radiation known on the time of proposal:
Inverse Compton gammas, what became important for  colliders.
[4] F.R. Arutynian, V.A. Tumanian, Phys. Lett., 4(1963), 176.
[5] E.Feinberg, H. Primakoff, Interaction of Cosmic Ray Primaries with Sunlight and Starlight, Phys. Rev.
73(1948) 449-469.
Channeling
[6]
A.I. Akhiezer, V.F. Boldyshev, N.F. Shul’ga, To the Theory of Radiation of Channeled Particles, DAN
USSR, 1977, vol. 236, N4, pp. 830-833.
Helical undulator/wiggler
[7] D.F. Alferov, Yu.A. Bashmakov, E.G. Bessonov, Generation of Circularly Polarized Electromagnetic
Radiation, Sov. Phys. Tech. Phys., 1976, vol. 21, N11, p. 1408.
We considered in [2] the possibilities to obtain helical fields with the help of static
magnetic fields, circularly polarized electromagnetic waves and helical crystals.
We concluded, that the method with static helical wiggler/undulator is the only
practical one. It was implemented into VLEPP project (1979).
This method solved the problem of the target overheating also (by the way), as the power
deposition for typical case is ~100W only.
3
So the view of conversion system implemented into Linac looks as the following
Linac
Linac
Undulator
Gammas
Target area
FIGURE 3: The helical undulator converter insertion. Soft bends used to direct/redirect accelerated bunch to the
accelerator line. This section installed after 100 GeV.
Perturbation of spin, energy spread and emittance is small, so this insertion can be installed in
general linac after the beam reached ~100GeV and accelerate/decelerate further on.
INTERACTION OF GAMMAS WITH MEDIA
Cross section for positron production in Born approximation can be found in [8]
[8] W. Heitler, The Quantum theory of radiation, Oxford University Press, 1954.
A.I. Akhiezer, V.B. Berestetzki, Quantum electrodynamics, Moscow, Nauka, 1981.
When E , E , E  2 mc 2
d ( E , E )
A 
 4Z 2 r02 ln(183 / Z 1/3 )G ( E E ) 
G( E E ) ,
d( E / E )
N0 X0
A –is its atomic weight, N0  6.022  10 23 is the Avohadro number, X0 is a radiation length
  e2 / c  1 / 137 ,
X01  4r02
N0 2 183
Z ln( 1/3 )[ cm 2 / gramm ] ,
A
Z
G(x) in this case
2
x(1  x )
G ( x )  x 2  (1  x )2  x(1  x ) 
3
9 ln(183Z 1/3 )
x  E / E
4
FIGURE 4: The differential cross-section of the pair production
E d ( E , E )
as the function of x for
4Z 2 r02
dE
E , E  2mc 2 [8]. The numbers at the top of each curve is energy of incoming gammas/mc2. The curves for
E =6,10 are valid for any energy. Hatched area corresponds to collected particles.
There is no dependence of the incoming photon energy in this function, the ratio only. This is
sequence of the assumption E , E , E  2 mc 2
One can see that at the edges of energy are fewer particles. As the method requires collection of
particles near the top of the energy, this circumstance reduces the efficiency of positron
generation. In any case the higher energy is desirable from the point of conversion efficiency. For
example, the increasing the energy of incoming photos from 5 to 25 MeV yields increasing the
efficiency about 6 times.
POLARIZATION
Longitudinal polarization of the secondary particle is a function of its energy, E , E and the
polarization  2 of the incoming gamma can be expressed [3]







   2  f ( E , E )  n  g( E , E )  n      ,


n – directed along the initial direction of the gamma radiation, n –directed rectangular to it.
f  E
E 1  E ( 1  2 2 / 3)
x  (1  x )( 1  2 2 / 3)
 2 1
,
2
2
( E  E ) 1  2 E E / 3 ( x  (1  x )2 ) 1  2 x(1  x ) / 3
where  1  ln183Z 1/3  F(Z ) ,

F( )   2 
n 1
1
,
n(n   2 )
2
5
2  1 
1
, x  E / E .
6
The function f weakly depends on Z, Fig.5.
E /E
FIGURE 5: The longitudinal polarization of the positron created as a function of its fractional energy [9].
The source of gammas must generate them with the highest possible value  2 , as the polarization
 2 defines the longitudinal polarization of the particles generated.
Evidently, amount of positrons must be one to one at IP conversion
The level of polarization of the positrons created, can be described by convolution of Fig.4&Fig.5
E  max
d ( E , E )
 2 ( E ) f ( E , E )
dE

dE


E  cap
.   
d ( E , E )
 dE dE
where N  is the number of positrons in the energy interval from the maximal possible
E max  sE max  2mc2 to E  cap .
6
3. UNDULATOR RADIATION
The main requirements for the photon beam is the monochomaticity, sufficient flux, polarization
Radiation of particles moving on Sin-like trajectories (undulator radiation) satisfies to all of these
requirements
TYPES OF UNDULATORS
There are different types of undulators, such as static electric/magnetic field, electromagnetic
waves, including laser radiation [2].
Laser radiation was specially mentioned in [10], 1992 as an example. Formulas are the same.
[10] E.G. Bessonov, Some aspects of the theory and technology of the conversion systems of linear Colliders, 15th
International Conference on High Energy Accelerators, Hamburg, 1992, p.138.
E.G. Bessonov, Proc. 6th International Workshop on Linear Colliders LC95, March 27-31, 1995, vol.1, pp594602, Tsucuba, Japan.
Later, the way with laser radiation was manifested as a direct line for JLC [11, 12].
[11] T. Hirose, in Poceedings of International Workshop on Physics and experiments with Linear Colliders, Morioka,
Iwate, Japan, September 8-12, 1995, World Science, Singapore, 1996, pp.748-756.
[12] T. Okugi, et al., Jpn. J. Apl. Phys. 35 (1996) 3677.
Field in an undulator can be represented as the following [7]


2z 
2z
,
H ( z )  ex Hxm Cos
 ey Hym Sin
u
u
x, y -- are the transverse coordinates, z is the longitudinal one,  u is period of the undulator,
Hxm , Hym are the magnetic field amplitudes in corresponding directions
Motion:


 (t  )  {  xm Cost ,  ym Sint ,   ( z )m Cos2t  }

r (t  )  { x m Sint  , ym Cost  , ct   (zm )Sin2t  } ,
  2 c /  u ,
 xm  Hym / Hc ,
 ym  Hxm / Hc , ( z )m  (  2xm   2ym ) / 4 ,
xm  c xm /  ,
 m  (  2xm   2ym )1/2 ,
   (1   2m / 4) ,
zm  c( z )m / 2 ,
Hc  2mc 2 / e u  10700[ G  cm ] /  u [cm ] , t   t  R(t  ) / c is the time in the moment of
ym  c ym /  ,
radiation.
Elliptically polarized gammas can be obtained with so called elliptical wiggler [13].
[13] E.G. Bessonov, E.B. Gaskevich, About the Spontaneous and Induced Radiation of the particles on higher
harmonics in the undulator with elliptically polarized magnetic field, Brief reports on Physics FIAN, 1985, No. 8,
p.17-21.
Helical undulator (or wiggler) has Hxm  Hym  H . In this case the motion is circular in
transverse plane, (helix in 3D) and particles radiate EM waves having circular polarization.
Deflection parameter or the undulatority factor defined
K     H / Hc  eH  u / 2mc 2  93.4  H [Tesla] u [m] .
7
Frequency of spontaneous radiation 
n 

n
n
n
n
 

2 

2
1  n 1   Cos 1    (1   )
1   1  2  (1  )
2

2
n

 
 
1    1 

2
2 

2
2

2 n 2

1  K 2   2 2
 n max
 2 2
1
1 K2
n –is the number of the harmonic,  is the observation angle, calculated from the forward
direction ,  n max  2n 2 / (1  K 2 ) corresponding forward direction.
So one can see that there are two ways to get high-energy photons following from the last formula
4nc 2
2.48  n
  
 n 

  5  [ MeV ] .
2
2 2
2
2 2
(1  K    )u (1  K    )u [cm]  10 
2
For 50 GeV, period u  1cm , 1  2.48[MeV ] .
Increase the energy of primary particles ~  2
Decrease the period of undulator ~ 1 / u
Electromagnetic field
These expressions give the possibility to calculate the electromagnetic field, radiated by the
particle

  

e(n  (( n   )   ))
E (t ) 

tt  R ( t ) / c ,
cR(1  n )3

where n is unit vector in the direction of observation and all values are taken at the moment of
radiation. Spectral angular distribution of the energy emitted by the particle on the area dS  Rd
is determined by expression
 2
 2
 c E ,
S



1
where E 
 E(t )exp(it )dt is a Fourier image of the electric field as a function of the time
2 
of observation t .
Parameter s
s
n

 n max
1
1
  (1  K 2 )(1  s) / s
 2 2
1 K2
Solid angle d  2  Sin  d  d 2  
 (1  K 2 )
ds
s 2 2
This angle does not depend on the harmonic number
8
Spectral/angular distribution [7]
 (1  K 2 ) dNn
K2

 4nM
Fn ( K, s) ,
ds
s2 2
d
1  K2
dNn
 2n 
1 K2
K
2 s  1 Jn ( n ) Jn ( n )
,
s(1  s )
Fn ( K , s )
where   2 K s(1  s) / (1  K 2 ) , Fn ( K , s)  Jn 2 (n ) 
1  K 2 (2s  1)2 2
Jn (n ) ,
4 K 2 s(1  s)
Jn and Jn are the Bessel function and its derivative,
M –is the number of the wiggler periods,
  e2 / c  1 / 137 is a fine structure constant.
In dipole approximation K  1 [14], ( s   n /  n max )
[14] E.G. Bessonov, A.A. Mikhailichenko, Some aspects of Undulator radiation forming for
conversion system of the linear collider, Preprint INP 92-43, Novosibirsk, 1992.
dN
ds
F1 ( K , s) 

n
dN n
ds
 4M
1
(1  2s  2s 2 ) ,
2
As a function of the angle
F1 ( ) 
1   2 2
,
2(1   2 2 )2
K2
1 K2
 nF ( K , s)
n
n
F2 ( K, s)  2s(1  s)(1  s  2s 2 )K 2
F2 ( )  2( K )2
Polarization in dipole approximation becomes
2s  1
 21   22 
or
1  2s  2s 2
1   4 4
.
(1   2 2 ) 4
 21   22 
1   4 4
1   4 4
Polarization becomes linear (  21   22  0 ), when the angle of observation   1 /  .
Total number of the photons from s = 1 (straight forward direction), to the threshold value s  st
defined by the maximal possible angle of incoming radiation, selected by the diaphragm
 t  (1  K 2 )(1  st ) / st .
The number of the photons
1
N n ( K , st )  
st
K2
ds  4nM
ds
1 K2
dN n
1
K2
s Fn ( K , st )  4nM 1  K 2 n ( K , st ) ,
t
9
In approximation   2K s(1  s) / (1  K 2 )  1 ( K  1 or/and   1 ) for harmonics with the
numbers n  1,2
1
K2
1 ( K , s )  (1  st )( 2  st  2 st2 )  ... ,  2 ( K , s) 
(1  st ) 2 (1  2st  2st2  4st3 )  ... .
2
6
10(1  K )

what is a function of the fractional energy only .
FIGURE 6: The density of the photons for two different energies of the primary beam.
Photon spectrum density, normalized to the maximal photon energy s  n / nmax
For harmonics n = 1,2,... in approximation K  1 or/and   1
1
2
 2 (1  2 s  2 s ), n  1

dNn
K2
2 s(1  s)(1  s  2 s2 ), n  2
 4nM


ds
1 K2 
...


 Fn ( K , s)
It is not a function of energy of primary electron beam
But the phonon flux as a function of (not normalized) energy is
dN n
d ( n / nmax )

dN n
dn

4nM
nmax
K2
4nM 2
K Fn ( K , s )
2 Fn ( K , s ) 
1 K
2 2 
10

The number of the photons
1
N n ( K , st )  
st
1
2
 6 (1  st )( 2  st  2st ), n  1

dNn
K2
K2

ds  4  nM

(1  st ) 2 (1  2st  2st2  4st3 ), n  2 ,

2
2
ds
1  K 10  (1  K )
...


where fractional energy st defined by diameter of the diaphragm hole
1
,
st 
 2d 2
1 2
R (1  K 2 )
where R is the distance to the target. If the diaphragm is large, d  R , then the number of the
photons is not a function of the beam energy at all. One can see, that for keeping the same
fractional energy, it is necessary to keep the ratio d / R constant. So reduction of the photon
density ~ 1 /  2 is not a limitation.
With increasing the energy, the cross section of the pair creation also increased, proportionally to
the energy of the photons (what is not a case for usual positron production)..
Restrictions:
2 2
Lowering the energy of the primary beam ~ N n    N n 
is not a serious problem.
1 K 2
Increased the energy spread in the primary beam E  N n   requires attention in each case.
Now one must collect particles from wider absolute energy interval now. Natural solution for this
is flux concentrator.
Energy acceptance in a damping ring is limited also. Energy compressor can help to avoid this
problem.
In general energy~150 GeV is about to be optimal one.
11
Descriptions of radiation, which represents a wiggler field as a wave
As we could see the number of photons, radiated at first harmonic by every particle, could be
written as the following
2
L K2
L e 2 H 2u2
2 H
,
N  4

4


Lr
0
u 1  K 2
u 4 2 m 2c 4

where L –is the length of undulator,   c / u , and we substitute here the K value.
H2
H2
can be treated as the photon density, n 
. So the formula can be rewritten as


N  L  n , where    r02 . This introduces the length of interaction as usual l  1 /   n , so the
The term
number of photons per initial particle is simply N  L / l .


Let 0 , k0 be the energy and momentum of incoming photon,  , k  –the energy and
momentum of the backward scattered gamma-quanta. Then these parameters linked by the
Compton formula [8]
v
v
  
v
0 (1  Cos )   (1  Cos )  02 (1  Cosˆ) ,
c
c
mc 
c

where v  v and mc 2 are initial velocity and the energy of incoming electron,  and   are




the angles between v and wave vectors k0 and k  respectively,  is the angle between k0 and

k  , see Fig. 7.
FIGURE 7: Cinematic of the Compton scattering.

Introducing the angle  between the line of incoming photon and the vector v as it is shown on
the Fig. 7, and supposing that    1 /  , one can obtain the energy of the backward scattered
photon

 max
mc 2  X  Cos 2 (  / 2)
,
  

2
2
2
1      XCos (  / 2)
 2  2
1
1  X  Cos 2 (  / 2)
where
4 2  0  Cos 2 (  / 2)
x
4 20 40

,
,
X




 mc 2
max
2
2
2
mc 
mc
1 x
1  X  Cos (  / 2)
x  X  Cos 2 (  / 2) . One can see, that dimensionless parameter X is proportional to the ratio of the
photon energy in the electron rest frame to the electron’s rest energy. In the case of wiggler
   / 2 , and x=X/2.
12
Angular spread in the beam
Let us compare this angle with the maximal angular spread of the particles in the beam. The last is
given by expression  m   /  u , where  is a normalized emittance,  u is an envelope
function value in the wiggler. As the envelope function is of the order of the wiggler length
 u  100 m , then for   104 cm  rad ,   4  105 (200 GeV), 1 /   2.5  10 6 , one can estimate
 m  104 / 4  109  16
.  107 , so  m  0.06 . Thus there is no input to the photon flux on the
target due to the angular spread in the beam. The beam dimensions in the wiggler will be of the
order r   u /   10 4  10 4 / 4  10 5  16
.  10 3 cm . The 10 criteria gives 10  r  0.016cm
or 0.16 mm, what gives the idea about possible aperture of the wiggler and also an influence of the
field inhomogeneties across the aperture.
 m   /  u
where  is a normalized emittance,  u is an envelope function value in the wiggler
Example :  u   u  M  100 m ,   104 cm  rad ,   4  105 (200 GeV), 1 /   2.5  10 6
 m  104 / 4  109  16
.  107 ,
 m  0.06
so
Angular spread does not affect the angular distribution
The beam dimensions in the wiggler
r   u /   10 4  10 4 / 4  10 5  16
.  10 3 cm
The 10 criteria gives 10  r  0.016cm or 0.16 mm, what gives the idea about possible aperture
of the wiggler and also an influence of the field non-homogeneity across the aperture.
Polarization for different harmonics as a function of angle and fractional energy. Averaged
polarization.
2s  1
 21   22 
1  2s  2s 2
An averaged value of circularly polarization of the photons concentrated in the solid angle
between 0 and  t  (1  K 2 )(1  st ) / st can be evaluated as
1
  2 n 
  2 n (s)
st
1

st
dNn
ds
ds
dNn
ds
ds
1


st
2n
(s)
dNn
ds
ds
Nn
.
Substitute here the expressions for  2 n , one can obtain in approximation K 2  1
13
  21 
3st
5st
,   22 
.
2  st  2st
1  2st  2st2  4st3
st  0 –absence of any selection
st  1 – straight forward direction
st  0.8 – selection in 20% down from the maximal possible energy of the quanta
If system collects only 20% of maximal possible energy down from the maximum, i.e. st  0.8 ,
then
  21  0.96 ,   22  0.95 .
For st  0.7 (30% interval)   21  0.92 ,   22  0.89
These figures indicate that the level of polarization is high.
About angular separation
The corresponding maximal values of the angles for selection (minimal value is zero for the
forward direction) are
0.65 1  K 2
0.5 1  K 2
2
and  ( st  0.8) 
.
 (st  0.7)  (1  K )(1  s) / s 


If the distance L between the end of helical wiggler and the target is L  200 m (distance between
the end of helical wiggler and the target),   4  105 (200 GeV), 1 /   2.5  10 6 , K 2  0.25 ,
st  0.8 , then corresponding diameter of the diaphragm at the face of target will be
2rD  L  s  5.6 10 2 cm  0.56mm .
For preparing the gamma flux of maximal possible polarization, the angular separation is
necessary. That formally gives the same threshold parameter in description of mean level
polarization of the flux. Other purpose of diaphragm is a protection of target
We have
2
FIGURE 8: Photon density and polarization for different harmonics. Hatched areas correspond to the angle 1  K
3
14
4. EFFICIENCY
Analytical calculations
We could see, that the number of photons N1 ( K )  4 
K2
, what is basically a reflection
u 1  K 2
L
of that fact, that on the formation length the  photons are radiated. The formation length for the
undulator is  f  2 2  u , so the factor L / u is the number of formation lengths at the
undulator’s length. Factor K 2 /(1  K 2 ) reflects the details of interaction (Landau-PomeranchukMigdal effect). In our case K 2  0.3 , so K 2 /(1  K 2 )  0.23  1/ 4 , so the number of quants goes
to
N1  
L
u
.
For 100m-long wiggler with period u  1cm , L /u  10 4 , N1  
L
u
 230 in all spectrum.
Preliminary estimations
Total cross-section per one atom
 tot  
1
0
A
7 A
.
G( x )dx 
N0 X0
9 N 0 X0
The number of the atoms N in the volume d  1cm 2
g[ g / cm3 ]  1cm 2  d[cm]
A[ g]
g –is the specific weight of the target material
N  N0
The number of the positrons at the exit of the target
N  N  tot N 
7
gd 7
N
 N 
9
X0 9
gd
–is the target thickness (length), measured as a fraction of the radiation length
X0
d – is the thickness of the target

.
Let 1/5 of all positrons only carrying the necessary level of polarization,   0.5
7 1
N / N    0.5  0.077 , or 7.7%
9 5
This estimation looks very close to that obtained from numerical calculation
We supposed also, that the phase volume of the positrons created, corresponds mostly to multi
scattering in a target, and the particles could be collected by appropriate focusing system.
15
For obtaining the formula, describing the spectrum of created positrons, we can write
d ( E , E ) d 2 N
d 2 N
1

dE dS ,
dE d  tot 
dE
dE dS
d 2 N
d 2 N

where
is the spectral density of the photon source, illuminating the target,
dE dS dE R2 d
dS  R2 d , d is the solid angle, R is the distance from the source to the target.
Differential cross section referred to the radiation length unit can be represented as the following
7 dE
, so
d ( E , E ) 
9 dE
dN 
K 2 L 7
 0.4 2
(1  E / E 1 )(1  e 7 / 9 )
dE
 c 9
For E0=150 GeV, L=150 m, K2=0.1,   0.5 (rad units)
1 dN 
 0.2 [1 / MeV ] .
N tot dE
More detailed analytical calculations take into account real geometry of the system.
zf
Lu  M u
z0
 f   f
zi
r
 i   i
Undulator
Target
z
FIGURE 9: Realistic geometry of conversion system.
First, the number of radiated photons goes [14]
N 1 
4K 2
 u (1  K 2 )3/ 2
 1 r 2
m

2
1

K2

 1 1
5 3rm4
4 K2   1 1  

  
1


  
2 3/ 2
5 1  K 2   z 3f zi3  
 z f zi  24(1  K ) 
where rm is the radius of the target (or the radius of the diaphragm installed before the target).
Taking into account that spectral angular distribution of the gammas from undulator has a form
for M  1,
d 2 Nn
1 d 2 n
1
1  n


 (   n ( )) .
2
dE dS En dE dS  n ( ) R ( ) 
For positrons
d 2 N
1

out
dE
 tot

d ( E , E ) d 2 N
7
 exp(  )W ( E , Eout ,    )ddE dE dS
dE
dE dS
9
16
7
where the factor exp(   ) reflects the photon flux attenuation by the target and function W
9
describes the probability WdE that the positron, created by the photon at the depth  with
initial energy E , will have the energy in the interval from Eout to Eout  dEout at the output of the
target [48]
[48] B. Rossi, “High Energy Particles”, N/Y, 1982.
W ( E , E ,    )dE
out

out

dE out  E 
    ln out

E  E 
(    )/ln 2 1
 
 
,
 ln 2 

where  ( x )   t x1e t dt is the Gamma function
0
 2 mc 2  E cap
The number of the positrons [10,14] in the energy interval E cap  Emax
n
created by the undulator radiation on the n-th harmonic
N  n ( E , E
out

max

Yˆ 
1
)
K 2
rm
c ln( 183Z 1 / 3 ) 0
si
dr 
sf
E max
Fn ds 
G( E , Emax )Yˆ ( E , Eout )dE ,

s(1  s) Ecap

E
7
ln 2
out
out
dE
exp(


)
W
(
E
,
E
,



)
d


(1   / ln 2 ) ,





 E out
9
 ln 
0

E  Eout
where   
. For thin target Y  1 , however. Finally
E
N n 
where  cap 
E  2mc
,
st E  2mc2
cap

max
K2 zf
ln( 183Z 1 / 3 ) 1  K 2 zi

1
 G( )d ,
ca p
1
2
periods , rm  r   z f
 2 MEmax 
 G( x)dx  0.773    0.681
1  K2

2
 0.454 3 , M is the number of
, so  is a fraction of what is the target radius with respect to the
size of the gamma spot at the target distance. For the first harmonics,
K2 zf
N 1  2 102  2 M
(1   cap )
1  K 2 zi
For   1 / 2, M  10 4 ,   0.2, K  1, z f  M u  2 zi ,  cap  0.7 , N1  3
Formula for average polarization
E max

  
 
1
 2 ( E ) f ( E , E )
d ( E , E ) d 2 N
E cap


7
 
exp(   ) exp( 
)dE ddE dS
dE dS
9
3X 0
dE
d ( E , E ) d 2 N
dE
7
exp(   )dE ddE dS
dE dS
9
17
.
Numerical codes for gamma-positron production
 LPI (KONN) was the first one dedicated to the polarized positron production.
FIGURE 10: Parameters of the code used for optimization.
dEn
6 2 n 2 Fn ( K ,  )
 Etot
 Etot  Fn ( K , ) ,
do

1  K 2   2 2



E tot 
r
4 
4
M e 2 K 2 2 
mc 2 o K 2  2
3 c
3
u
PROBABILITY
2

n

n
Fn  F  F ,

1
1  K 2   2 2
F ( K , )   J n (n ) 
J n (n )  ,
2
2 K


n
2
 1  K 2   2 2  2
2

 J n (n ) , where   2 K /(1  K 2   2 2 )
Fn ( K , )  J n (n )  
2 K


dE
dEn
2
n ( K ,  ) d  Etot
o do do  n o do do  Etot n  Fn ( K , )do  Etot n F

1
Probability to have a certain polarization
2n 
Fn  Fn 1  K 2   2 2 J n (n ) J n (n )


Fn
K
Fn ( K , )
Solenoid
Target
Be windovs
Lithium
Accelerator structure
FIGURE 11: Optimized collection and separation system. Particles with low energy are overfocused. Dimensions are
in mm.
18
This selection system used a lithium lens and a diaphragm as energy separator: the particles with
lower energy became overfocused.
1 dN 
 0.2 MeV 1 , x pc  2 MeV  cm , E  6 meV .
N tot dE
 UNIMOD2 (an analog of EGS type code)
Codes for calculation the efficiency of the photon interaction with media. First of all there are the
codes used for modeling the high-energy physics phenomena, for example EGS type codes. The
output file of UNIMOD2 (an analog of EGS) is used by the code CONVER [16] for rather fast
calculations with the targets having a different size and form.
[16] A.D. Bukin, A.A. Mikhailichenko, Optimized Target strategy for Polarized Electrons/Positrons production for
Linear Collider, Budker INP 92-76, Novosibirsk, 1992.
Individual history of about 6000-10000 incoming photons (depending of the accuracy required)
OBRA  PARMELA for furthers transport.
The main result of these considerations that the efficiency of the particle
production could be made ~6% for each initial photon. The mean polarization can
be up to 65-70% for the reasonable length of the wiggler ~100m.
19
5.UNDULATOR DESIGN
Undulator proposed by V.L. Ginsburg [17]. H. Motz referred to him in his practical design of his
wiggler [18].
[17] V.L. Ginzburg, To the radiation of microwaves and its absorption in air, Izv. AN USSR, ser. Phys. 1947, vol. 11,
N2, pp. 165-181.
[18] H. Motz, Applications of Radiation from fast Electron beams, J. Appl. Phys., 1951, Vol 22, N5, pp. 527-535.
Helical undulator is a bifilar helix with opposed currents.
Technical proposition for helical field generation with helical wiring was made in [19].
[19] R.C. Wingerson, “Corkscrew” -a Device for Changing the Magnetic Moment of Charged
Particles in a Magnetic Field, Phys. Rev. Lett., 1961, Vol. 6, No. 9, pp. 446-449.
FIGURE 12: Schematics of the helical dipole wiggler.
This is a bifilar helix with currents opposed. This type is broadly in use now.
Helical undulators with permanent magnets and short period described in [20]. The cost is the
point of concern only.
[20] P. Vobly, Helical undulator for Production circularly Polarized Photons, In Workshop on New Kinds of
Positron Sources for Linear Colliders, SLAC, Marc 4-7, 1997, SLAC-R-502. pp.429-430.
Micropole wigglers described in [21].
[21] P. L. Cshonka, NIM A, Vol. 345, No 1, 1994.
The progress in design of short period wigglers with high field one can find in [22-24].
[22] A.D. Cherniakin et al., The development of the conversion system for VLEPP project, Proc. of the 12 Int.
Conference on high energy Accelerators, Batavia, 1983, p.131.
[23] A.A. Anashin et al., Superconducting helical undulator for measurements of polarization of interacting beams in
VEPP-2M storage ring , Preprint INP 84-11, Novosibirsk, 1984.
In [24] the results of calculations and testing the models with the period 0.6 and 1-cm are
represented.
[24] T.A. Vsevolojskaya, A.A. Mikhailichenko, E.A. Perevedentsev, G.I. Silvestrov, A.D. Cherniakin, Helical
Undulator for conversion system of the VLEPP project , XIII International Conference on High Energy
accelerators, August 7-11, 1986, Novosibirsk.
Each undulator requires careful optimization.
Lot of 3D codes allowing computation of helical fields are on the market now, [25] for example
[25] A.N. Dubrovin, E.A. Simonov, MERMAID- MEsh-oriented Routine for Magnet Interactive Design, User’s
guide, Novosibirsk, INP, 1992.
Sometimes they are the codes are the same as for calculation of 2D fields, but with substitution of
coordinate dependence
  x  iy  e i    exp( iz /  u ) ,
what is, basically, the twist with the wiggler period.
20
Tested undulators with period 6 and 10 mm
Superconducting undulator
Period
– 1.0cm
Axis field ~ 5 kG
K  0.47
Length ~ 30 cm
Current in each of 22/turn coil ~ 200 A
This undulator was supplied with the captured flux also
That was made with the help of superconducting transformer
FIGURE 13: The 30 cm long superconducting undulator with period 10 mm and the axis field ~ 5 kG.
FIGURE 14: The 30 cm long superconducting undulator details. At the left: SC transformer and SC
switches allow capturing the flux.
Modular design was chosen for superconducting undulator. Sections~5 meters long.
21
Pulsed undulator
Period – 0.6 cm
Axis field ~ 6kG
K  0.35
Length ~ 1m
Current
~10 kA
Pulse duration ~50  sec
Voltage
~ 1.19 kV
Inductance
– 1.3 H
Repetition rate –25Hz
FIGURE 15: The pulsed wiggler. Details of cooling system are visible at the left picture.
22
6. TECHNICAL DETAILS OF COLLECTION SYSTEM
Energy deposition in a target
Special considerations have done to estimate the energy deposition in the material of the target. It
was found that this value is around 250 MeV/g at the end of the target. The thickness of the target
was about 0.2 cm. This yields the temperature gain of the order 116 deg for the beam with 1010
positrons in the bunch.
Dulongue-Petit law. Titanium target [26]
[26] J.Rossbach, K. Floetmann, A High Intensity Positron source for Linear Collider, DESY, DESY M-91-11
3
k B NT  rotation  3k B N AT
kB  1.38  10 23 Joules / oK
2
Q
Heat capacity cp 
 25 Joule / moll / deg
T
Q
FIGURE 16: Efficiency as a function of the thickness for Ti target. Top curve for gammas 25 MeV in full energy
spectrum. The rest curves for the upper half of energy spectrum.
Lithium lens (mentioned above)
[27]
G.I. Silvestrov, Problems of production high intensity beams of secondary particles, XIII International
Conference on High Energy Accelerators, August 7-11, 1986, Novosibirsk.
FIGURE 17: Lithium lens concept.
23
Solenoidal lens
FIGURE 18: Positron collection system with partial flux concentrator, developed for CESR (recently doubled
positron production).
Diaphragm
Diaphragm required for target for angular separation of the gammas. It also required for helical
undulator protection.
One of the kinds considered in [28]
[28] A.D. Bukin, A.A. Mikhailichenko, Optimized Target strategy for Polarized Electrons/Positrons production for
Linear Collider, Budker INP 92-76, Novosibirsk, 1992.
Berillium
Iron
Iron
Target
FIGURE 19 : The diaphragm schematics. DC current magnetizes iron.
24
7. PERTURBATIONS
Emittance perturbation in an undulator
d x , y
  d ( E / E )

  Hx , y  x2, y 
dz
 
dz

H(z) 
1  2
1

  ( z )  (    )2  ,

 (z) 
2
z—is a longitudinal coordinate here.
A -means averaging over period of the undulator.
A - means averaging over spectrum of radiation.

K u

Sin
z 2
z

Sin ,
u 
u
Dispersion invariant becomes

 ( z ) 

H   2 
Cos
z
,
u
 u  2

2
Cos2
2
dN
  max 
 E 
 E  dN

dE  
ds ,
  s2
    
 E 
 E
 E  dE
ds
2

2
s

,
K
z
u
E
1

E max
1
 2 2
1 K2
We have
dN
ds

n
K2
 4M
ds
1 K2
dN n
 nF ( K , s)
n
n
M—is the number of periods
F1 ( K , s) 
1
(1  2s  2s 2 ) ,
2
1
7
0 s (1  2s  2s )  30 ,
2
2
F2 ( K, s)  2s(1  s)(1  s  2s 2 )K 2 ,
1
13
 s (1  s)(1  2s  2s )  420
3
2
0
2
2
K 2  2 2  
K2
7 
K 4  r0 
7
 

2

M


  2M 

2 
2
2
2 3 
2  (1  K ) E 
1  K 30 2 (1  K )   u 
30
r0  e2 / mc 2
Emittance perturbation in undulator is negligible
25
Spin perturbation in an undulator
This topic was described in [29]
[29] E.A. Perevedentsev, V.I. Ptitsin, Yu.M. Shatunov, Spin behavior in Helical Undulator. In *Hamburg 1992,
Proceedings of 15th Int. Conf. on High-energy accelerators, vol. 1* 170-172. (Int. J. Mod. Phys. A, Proc. Suppl.
2A (1993) 170-172).

Vector P (spin vector) is defined in the rest frame of the positron or electron.
The fields are defined in the laboratory system.
  

  
dP
e 
g  2   
g  2 ( B)(  P)  g  2
1  P  (E   )


 

1  
 ( P  B)  (  1) 

dt m 
2 
2
2
  1
c
 2

dP  
 P  s
dt
 
 
 v/c ;
E , B -- electric and magnetic fields in the laboratory frame
g2

 1159652
.
 10 3 
2
2

c
Coming to rotating system of reference  
u
FIGURE 20: Representation of the spin motion in rotating system of reference.


dP  
 P  (  s  )
dt


Components of the vector  eff   s   :

 e 
g  2
c
eff  
1


H
;
0
;



2 
u 
 mc 




g  2  eH  u c
c  
g  2 K c 
c  
eff      1  

 ; 0;
 e ; 0;
e 
  1  


2  mc    u
 u  
2   u
u 

  depend on energy ~
K

26
Spin frequency is

c
s 
u
2
2
1   g  2  K  1      (1  1 1   g  2  K )   
s

2   2
2 
2   2
2
2
g  2 K2
   (1  
)
 2  2
2
Direction of rotation







 
g  2 K 


n
 1  

e
;
0
;
e


2  
s


g  2  K2
 10 6 K 2
2

2  

Depolarization time due to radiation in an undulator due to spin-flip processes is, as always
~  /  2 r0 bigger, those time constant of energy losses. The relative energy losses of particles in
undulator are 0.001-0.005 however. Additional factor ~2/9 is working in favor of longitudinal
polarization.
Spin perturbation at the interaction point
Due to huge magnetic field of incoming beam the vector of spin rotates at the angle
  2  E[GeV ] / 0.4406 with respect to the vector of momenta. This effect was a point of
concern from the very beginning, 1979.
The effect yields a lost of a few percent of polarization. The mostly complete known description
one can find in [30]
Possible depolarization
P  1  
2
[30] E.A. Kushnirenko, A. A. Likhoded, M.V. Shevlyagin, Depolarization Effects for Collisions of Polarized
e  e  beams, IHEP 93-131, SW 9430, Protvino 1993.
Resistive instability in an undulator chamber
The resistive wall instability for the beam, moving in the vacuum chamber of the wiggler
considered in [31].
[31] A.A. Mikhailichenko, V.V. Parkhomchuk, Transverse Resistive Instability of a single Bunch in a Linear Collider,
Preprint INP 91-55, Novosibirsk, 1991.
Heating

c
l
 mm waves !  l is the bunch length
Skin depth  

2
0 
1
for room temperature Cu; 0  4  10 7 H / m
17
.  10 Ohm  m
8
27
Area   2 a   , a is a radius of the vacuum chamber
Resistance for all 100 meters,  l  1mm
0 
0  c 100  1.7  10 8 4  10 7 3  108
L
L
L

 300[]
R


   2 a 
2
2 a  2 l
2 3  10 3
2  1.7  10 8  10 3
Number of the particles N  1010 , peak current for  l  1mm
I 
eN
1.6  10 19  1010

 480 A
( l / c) 0.001 / 3  108
Pulsed power for single bunch per second for all 100 meters for  l  1mm
(eN )2
0.001
P   I 2 Rdt 
 R  ( l / c)  4802  300 
 2.3  104 [Watt ]
2
( l / c)
3  108
One can easily scale this number to the necessary repetition rate and to the number of the bunches
per train and for other  l and N. (For  l  0 .1mm the losses will be 30 times higher,
P  7  10 3 W ). Magnetic field on the inner surface is
0 .4   I
H
 300 G .
2a
Liquid helium temperature  drastic reduction of resistance
This looks as a weak requirement
Wall irradiation
Angle  
3
r 3  10
  4  105  3  105  12 , For beam with E=200GeV

 3  105 ,
L
100
n2 2 
2.48  ( / 105 )2
2.48  16
E max 

[ MeV ] 
 0.27 MeV
2
2 2
2
2 2
1 K   
(1  K    )[cm]
144  1
Quadratic dependence on inner diameter.
The wall illumination was considered and looks as a weak requirement.
Depolarization in a target
The change of  , when the particle goes from the point of creation  to the output surface of
the target is described by the length of depolarization ldep  3 X 0 so

   exp( 
 
)
3X0
Thickness of the target   X0 / 2 , so radiative depolarization in the target after creation is about
8.3% , as
1
1
exp(
)  1
 0.917 ,
223
12
out
where additional factor ½ reflects the mean path length of the individual positron in the target.
Numerical calculations shows that the mean path length even less than ½ reflecting the total
tendency that the particles created at the out side of the target have more probability to come out
of the target. This effect included in calculations.
28
8. HOW TO REDUCE THE LENGTH OF THE UNDULATOR
Few targets in series
Attenuation coefficient k  exp(  79  ) for   0.25 is around 0.82 i.e. only 18% of photons is lost.
So the second target can be used in additional to the first one [32].
[32] A.A. Mikhailichenko, Dissertation, BINP, Novosibirsk, 1986.
This allows further utilization of the gamma –beam, passed through a thin converter. Also helps in
reduction of target heating. Use the target thinner, than optimal is a way to go.
The combining scheme [32] based on the possibility to stack bunches in longitudinal phase
space.
FIGURE 21 : Multitarget arrangements [32]. Energy provided by acceleration structures A1 and A2 are slightly
different.
FIGURE 22: Isometric view of previous figure. The solenoidal capturing optics is shown at the right.
Here the gammas from undulator are coming from the left side and illuminate the target T . The
focusing lens L collects the particles and adjusts for further optics. An acceleration section A1
gives the energy E1 for the secondary beam. This beam of positrons is bend with the help of
magnet M1 . The magnet M1 and a part of the magnet M3 with quadrupoles l make an achromatic
parallel shift of the positron beam. The second acceleration section A2 , gives the lower energy 29
E2 to the beam, collected from the second target. The magnet M2 with the part of the magnet M3
and lenses, also adjusted for achromatic parallel shift of the beam with the energy E2 . The
difference in the path lengths of these two lines is an integer and a half of the wavelength of the
section A3 . This section eliminates the energy difference. D is the gamma beam dump.
Some details. At the end of the magnet M1,2
    (1  Cos ) , (s)  Sin ,   z / 
In the middle of the lens l  l    (1  Cos )  L  Sin
  100cm , then  l  100  0.134  150  163cm . If energy
E10=200MeV , E max  20 MeV and we collect half of this i.e. 15  5MeV , so E1=215  5 MeV ,
10
then radial displacement will be r   l 
 8cm (  4cm) .
200
( HR)
Focal distance of the lens is L 
. Supposing that the length of the lens is l=20cm, gradient
Gl
( HR) 667kGs  cm
must be G 

 0.17kGs / cm only.
lL
200cm  20cm
Energy of the second acceleration structure can be E2 =205  5MeV
This is a good place to arrange energy selection.
Optics could be realized easily. Reduction of the length ~ ½. 150 75metrs.
Let    / 6 ,
L=300cm ,
Form-factor for Ti target
An idea to use the heavy targets as a kind of a needle for positron production was under discussion
in many laboratories, including BINP since the very beginning of e  e  business. The result was
always negative – shower expands too fast.
This is not the case however for the positron production though the gammas form wiggler, what
was mentioned in 1997 in [29].
[29] A.A. Mikhailichenko, Use of undulators at High Energy to Produce Polarized Positrons and Electrons,
Workshop on New Kinds of Positron Sources for Linear Colliders, SLAC, Marc 4-7, 1997, SLAC-R-502.
Program for calculations described in [28].
[28] A.D. Bukin, A.A. Mikhailichenko, Optimized target strategy for polarized electrons and positrons production
for linear Collider, Budker INP 92-76, Novosibirsk, 1992.
FIGURE 23: The positrons created by gammas can escape from the target, as there is no size increase for gamma ray
flux. This is crucial difference with ordinary positron production.
Length/diameter  10. For E  20 MeV efficiency was found to be  2.67% for target of 4cm
long (compare with 1.7% for infinitely wide target),    0.1cm Diameter of the target 0.4cm.
Not possible for Tungsten target neither for electrons nor for gammas
30
9. PROPOSAL FOR SLAC ACCELERATOR
E~50GeV
N  5  1010
  3  103 cm  rad
Wiggler installed in crossover with  *
 ( z)   * 
z2
*
  * (1 
Beam size at the out of the undulator changes only 10%
z2
 *2

)
L2
 *2
 0.2
L=2m   *  450cm
The beam size  
 *
 3.67  10 3 cm . Aperture  10  0.37mm only

Undulator optimizations made allow to have K=0.2 ,  u  2mm for Inner diameter = 2mm.
E max
n2 2  2.48  ( / 105 )2


[ MeV ]  12.4 MeV
1 K2
(1  K 2 )[cm]
N  1  N e
4K 2
M
1 K 2

N 1
Ne 
 2.3
Analytical estimation
N   2 102 s   M
2
K2 zf
  (1  1 )
1  K 2 zi
(9.1)
z f – z coordinate of the end of the undulator
zi – z coordinate of the beginning of the undulator ( zi > z f )
1 –energy range of collected positrons ( 1 =0 ==> full energy interval)
 s – fraction of  (1/3 for s=0.9)
 – thickness of the target
zf
Lu  M u
z0
zi
Undulator
Target
FIGURE 24: Geometrical parameters defined by formula (9.1).
For  s =1/3,   015
. , K=0.2 , 1  0.9 , z f / zi  1 , M  400 / 0.2  2 103
K2 zf
N   2 10  s 
  (1  1 ) =0.023 (compare with 0.025)
1  K 2 zi
2
2
31
Numerical calculation of efficiency for E  10 MeV in Tungsten.
E  7.5  2.5 MeV ; Thickness   015
.
Angle of capture 0.5rad  Efficiency
N N N

 0.014  2.3  0.025
N N N
What gives N  0.025  5  1010  1.25  109
For B-Factory proposal was made [33] on how to generate the polarized positrons in amount
which can satisfy the B-Factory.
[33] A.A. Mikhailichenko, The feasibility of Polarized e+ e- collisions AT SLAC B factory, (Cornell U., LNS). CLNS99-1645, Jun 1999. 12pp.
Undulator Converter Test Facility
It is a good idea to make a test facility at SLAC for the purposes of testing technology of helical
wiggler production and methods of diagnostics.
FIGURE 25: Possible location of the test facility.
32
10. CONCLUSION
Polarization is a powerful tool for high-energy physics, beyond 100 GeV especially, when
electromagnetic and weak interactions become comparable.
Polarized positron generation for linear collider application can be arranged with the help
of conversion of circularly polarized gammas in media.
Optimal yield of polarized positrons can be obtained with thin (in terms of X0) converter.
Required number of circularly polarized gammas can be obtained from initial particles
with the help of helical undulator.
Conversion efficiency for polarized positrons can be made one to one, referred to IP.
Considerations and modeling (including manufacturing prototypes) done show, that
technical components of conversion system, such as undulator, target, collection optics
can be build with no doubt.
Spin handling strategy for all collider is well known also.
The scheme eliminates target overheating, as the power dissipation is ~100W.
SC undulator is relatively simple and inexpensive.
The length could be reduced by some strategy:
Length
Few targets
Form-factor
~100m

~50m

~25m
System starts working from energy ~100 GeV with undulator having period ~ 1cm.
Perturbation of spin, energy spread and emittance is small, so this insertion can be
installed in general linac after the beam reached ~100GeV and accelerate/decelerate
further on.
High field flat wiggler allowing generation of unpolarized positrons.
This strategy can satisfy LC operations at earlier stages for high power beams (DESY
way).
Ordinary positron production satisfies this strategy also for low power.
The method could be tested at 50 GeV with specially optimized undulator with period
2mm.
Undulator Converter Test Facility at SLAC (UCTF) is a good way forward.
IT IS GARANTEED TECNICALLY THAT FUTURE LINEAR COLLIDER CAN HAVE
BOTH
e
AND
e  POLARIZED
33