Proposal for an Enhanced Optical Cooling System Test in an Electron
  Storage Ring
PROPOSAL FOR AN ENHANCED OPTICAL COOLING SYSTEM 
TEST IN AN ELECTRON STORAGE RING 
                     E.G.Bessonov, M.V.Gorbunkov, Lebedev Phys. Inst. RAS, Moscow, Russia,     
                              A.A.Mikhailichenko, Cornell University, Ithaca, NY, U.S.A. 
                                                                        Abstract  
    We are proposing to test experimentally the new idea of Enhanced Optical Cooling (EOC) 
in an electron storage ring. This experiment will confirm new fundamental processes in beam 
physics and will demonstrate new unique possibilities with this cooling technique. It will 
open important applications of EOC in nuclear physics, elementary particle physics and in 
Light Sources (LS) based on high brightness electron and ion beams. 
1. INTRODUCTION 
    Emittance and the number of stored particles –N in the beam determine the principal 
parameter of the beam, its Brightness what can be defined as / x z sB N γε γε γε= , where each , ,x z sγε  
stands for invariant emittance associated with corresponding coordinate. Beam cooling 
reduces the beam emittance (its size and the energy spread) in a storage ring and therefore 
improves its quality for experiments. All high-energy colliders and high-brilliance LS’s 
require intense cooling to reach extreme parameters. Several methods for the particle beam 
cooling are in hand now: (i) radiation cooling, (ii) electron cooling, (iii) stochastic cooling, 
(iv) optical stochastic cooling, (v) laser cooling, (vi) ionization cooling, and (vii) radiative 
(stimulated radiation) cooling [1-3]. Recently a new method of EOC was suggested [4-7] and 
in this proposal we discuss an experiment which might test this method in an existing 
electron storage ring having maximal energy ~ 2.5 GeV, and which can also function down to 
energies of ~100-200 MeV.  
Figure1: The scheme of the EOC of a particle beam (a) and unwrapped optical scheme (b) 
     EOC [4] appeared as the symbiosis of enhanced emittance exchange and Optical 
Stochastic Cooling (OSC) [8-10]. These ideas have not yet been demonstrated. At the same 
time the ordinary Stochastic Cooling (SC) is widely in use in proton and ion colliders. OSC 
and EOC extend the potential for fast cooling due to bandwidth. EOC can be successfully 
used in Large Hadron Collider (LHC) as well as in a planned muon collider.  
    The EOC in the simpiest case of two dimensional cooling in the longitudinal and 
transverse x-planes is based on one pickup and one or more kicker undulators located at a 
distance determined by the betatron phase advance betxψ = ,2 ( 1/ 2)p kkπ +  for first kicker 
undulator and betxψ = ,2 k kkπ  for the next ones, where kij  = 0, 1, 2, 3,… is the whole numbers. 
Other elements of the cooling system are the optical amplifier (typically Optical Parametric 
Amplifier i.e. OPA), optical filters, optical lenses, movable screen(s) and optical line with 
variable time delay (see Fig.1). An optical delay line can be used together with (or in some 
cases without) isochronous pass-way between undulators to keep the phases of particles such 
that the kicker undulator decelerates the particles during the process of cooling [6], [7].  
2. TO THE FOUNDATIONS OF ENHANCED OPTICAL COOLING  
    The total amount of energy carried out by undulator radiation (UR) emitted by electrons 
traversing an undulator, according to classical electrodynamics, is given by 
      
2 2 2 22
to t e uE r B Lβ γ= ,                                                 (1) 
where  is the classical electron radius;  e,  are the electron charge and mass 
respectively; 
2 /er e m c=
2B  is an averaged square of magnetic field along the undulator period 
uλ ; 
/v cβ = is the relative velocity of the electron;  is the relativistic factor; 2/ eE m cγ = u uL Mλ=  is 
the length of the undulator; and M is the number of undulator periods.  For a planar harmonic 
undulator 2 20 / 2B B= , where B0 is the peak of the undulator field. For a helical undulator 
0B B=
.  The spectral distribution of the first harmonic of UR for  M>>1 is given by [11]  
              1 1/ ( ) (0
cl cldE d E fξ ξ ξ= ≤ ≤ ,                                           (2) 
where , , 2 21 /(1 )cl cltotE E K= + )221(3)( 2ξξξξ +−=f ξ = 1,min 1/λ λ , 1min 1 0|θλ λ == , , ( ) 1f dξ ξ =∫ K = 2 /ue B λ  
 is the deflection parameter, 22 em cπ
1 (1 ) / 2u K
2λ λ ϑ= + + γ  is the wavelength of the first harmonic 
of the UR, ϑ γθ= ; θ  is the azimuthtal angle between the vector of electron average velocity 
in the undulator and the undulator axis.   
    Electrons have effective resonant interaction in the field of the kicker undulator only with 
that part of their undulator radiation wavelets (URW) emitted in the pickup undulator if the 
frequency bands and the angles of the electron average velocities are selected in the ranges   
       ( )
∆⎛ ⎞ = ∆ =⎜ ⎟
⎝ ⎠ M
+                                              (3) 
nearby maximal frequency and to the axes of both pickup and kicker undulators. Optical 
filters which are tuned up to the maximal frequency of the first harmonic of the UR can be 
used for this selection. In this case screens must select the URWs emitted at angles 
( )URW Cϑ ϑ< ∆  to the pickup undulator axis both in horizontal and vertical directions before 
they enter optical amplifier (to do away with the unwanted part of URWs loading OPA). In 
this case the angle between the average electron velocity vector in the undulator and the 
undulator axis will be small:  
         ( ) ( )e Cϑ ϑ∆ < ∆ .                                                        (4) 
    Below we suggest that the optical system of EOC selects a portion of URWs, emitted in 
this range of angles and frequencies, by filters, diaphragms and/or screens. This condition 
limits the precision  of the phase advance  determined by the equation  ,
x zδψ ,
                         ,                                                        (5) ,( ) ( )
x z Cδθ θ< ∆
where  is the change of the angle between the electron 
average velocity and the axis of the kicker undulator owing to an error in the arrangement of 
, , , , ,( ) (2 / )sin( )
bet bet
x z x z bet x z bet x zAδθ π λ δψ= ,
undulators,  is the amplitude of the betatron oscillations of the electron in the storage 
ring, in the smooth approximation δψ ,  is the displacement of the kicker 
undulator from optimal position,  is the length of the period of betatron oscillations.  
, ,x z betA
, , ,2 /
x z x z betsπ λ= ∆ s∆
, ,x z betλ
   The number of the photons in the URW emitted by electrons in suitable cooling frequency 
and angular ranges (3) is defined by the following formula (see Appendix 1) 
              
1max 1
,                                                     (6) 
where , 2 21 1( / ) 3 / 2 (1 )
cl cl
totE dE d E M Kω ω∆ = ∆ = + 1maxω = 1min2 /cπ λ ,   [11]. 
Filtered URWs must be amplified and directed along the axis of the kicker undulator.  
2 1 137e cα = / ≅ /
   If the density of energy in the URWs has a Gaussian distribution with a waist size ,
w x zσ σ> , 
, the R.M.S. electric field strength / 2R uZ L> clwE  of the wavelet of the length 1min2Mλ  in the 
kicker undulator defined by the expression    
                            
2 2 3/ 2
M K M
σ λ σ
.                                          (7)  
where ,
x zσ  are the electron beam dimensions, 
, 1mi4 /R w cZ nπσ λ=  is the Rayleigh length. If 
x z w cσ σ< ,W w c, , the R.M.S. electric field strength clwE  of the wavelet becomes  σ σ=
                              
                                                           (8) 
where ,w cσ = 1min /8uL λ π  is the  waist size corresponding to the Rayleigh length .  /2R uZ L=
     Note that electric field values (7), (8) do not take into account quantum nature of emission 
of URWs in a pickup undulator. They are valid for . Such case can be realized only 
for heavy ions with atomic number  and for deflection parameter . If, according 
to classical electrodynamics, , then it means that in the reality, according to quantum 
theory, one photon is emitted with the energy 
1phN >>
10Z > 1K >
1<phN
max,1ω  and with the probability . 
In this case the electric field strength is determined by the replacement of the energy 
1em php N= <
clE∆  on 
 in (7) and the frequency of the emission of photons  
, where 
1 1 1,ph
q clE E N ω−∆ = ∆ ⋅ = max ph emf f p= ⋅ =
phf N f⋅ < f  is the revolution frequency of the electron in the storage ring.  
     If the number of electrons in the URW sample is , then URW emitted by an electron i  
in pickup undulator and amplified in OPA decrease the amplitudes of betatron and 
synchrotron oscillations of this electron in the kicker undulator. Other  electrons emit 
URWs including  non-synchronous (for the electron i ) photons, which are 
amplified by OPA and together with noice photons of the OPA increase the amplitudes of 
oscillations of the electron i . If the number of non-synchronous photons in the sample 
, where - is the number of noise photons in the URW 
sample at the amplifier front end [6], [7], then the jumps of the closed orbit and the electric 
field strengths are determined by the replacement of the energy 
,e sN
, 1e sN −
,( 1)e s phN N−
, ,( 1)ph e s ph nN N N 1Σ = − + <N nN
clE∆  on 1
qE∆  in (7), (8) and 
the frequency of the emission of photons is ,em phf f p f NΣ fΣ= ⋅ = ⋅ < . In the opposite case, 
, the electric field strengths are determined by the replacement of the energy , 1phN Σ > 1
clE∆  
on  in (7) and the frequency of the emission of photons is .  1 , 1
phE N ω
Σ∆ = ⋅ ,max
2 /e s e sN M N
     In our case , 1,min , ,0 , λ σΣ= eN Σ  stands for the number of electrons in the 
bunch, ,0sσ  is the initial length of the electron bunch.  
    The maximum rate of energy losses for the electron in the fields of the kicker undulators 
and amplified URW is  
max ( ) |
w w c
loss w u m ph kick amplP eE L f N N σ σβ α⊥ == − Φ =
1,min
8 ( )
ph kick ample f N N K
π π α
,                 (9) 
where γβ /K=⊥ ;  is the number of kicker undulators (it is supposed that electrons are 
decelerated in these undulators); and 
kickN
a m p lα  is the gain in the optical amplifier. The function 
1( ) | phph N em phN p NΦ = = , 1( ) 1phph NN >>Φ =  takes into account the quantum nature of 
the emission (the frequency of emission of photons ~ phf N⋅  and the electric field strength 
~1/ phN ). It follows, that quantum nature of the photon emission in undulators leads to the 
decrease of the maximum average rate of energy losses for electrons in the fields of the 
kicker undulator and amplified URW by the factor 1( ) | 9.phph N phN NΦ = 3 .  
     The damping times for the longitudinal and transverse degrees of freedom are  
         ,0, max
s EOC
lossP
,           
,0 ,0
, , max
x EOC s EOC
x loss x kick
σ β σ
= = ,                                     (10) 
where ,0Eσ  is the initial energy spread of the electron beam, Ploss  stands for the power losses 
(9), ,0xσ  is the initial radial beam dimension determined by betatron oscillations, , 0x kickη ≠  is 
the dispersion function in the kicker undulator, . Note that the damping 
time for the longitudinal direction does not depend on 
,0 , ,0( /x x kick E Eησ η β σ
,x kickη  and one for the transverse 
direction is inverse to ,x kickη . Factor 6 in (10) takes into account that the energy spread for 
cooling is ,02 Eσ , electrons does not interact with their URWs every turn (screening effect) and 
that the jumps of the electron energy and closed orbit in general case lead to lesser jumps of 
the amplitude of synchrotron and betatron oscillations [6].  
     The equilibrium spread in the positions of the closed orbits 2
, the spread of betatron 
amplitudes  2
A⎛ ⎞⎜ ⎟
 and corresponding beam dimensions ,EOC EOCx xησ σ  determined by EOC 
        
2 2 1
, , 1 ,
/ 2 / 2 | ( 1 / ) | |
2 2ph
EOC EOC
EOC EOC
x eq x eq x N e s n ph
eq eq
A x N N
N xη ησ σ δΣ >
⎛ ⎞ ⎛ ⎞= = = = − +⎜ ⎟ ⎜ ⎟
⎝ ⎠ ⎝ ⎠
,              (11) 
where 1 2 max( /x loss )x E Eηδ η β
−= ∆  is the jump of the electron closed orbit determined by the energy 
jump  of the electron in the fields of the kicker undulator and its amplified 
URW (corresponds to one-photon/mode or one-photon/sample at the amplifier front end).  
max max /loss loss phE P f N∆ =
     The equilibrium relative energy spread of the electron beam 
                                                                                                (12) 
EOC EOC
E eq x kick x kickE ησ β σ η= ,/
     Note that jumps of closed orbits 1xηδ ~1/ .  That is why the electron bunch dimensions 
(11) at the same number of particles in the sample and relativistic factor are much higher ions 
one. As a sequence of small electron charge the number of photons in the URWs 
, 87% of UWRs are empty of synchronous photons and every URW has 
 non-synchronous photons. That is why the contribution of noise photons for 
electrons is greater ( ) then for heavy ions.  
21.15 10 1phN
, 1phN Σ >
/ 87n phN N Nn
    The power transferred from the optical amplifier to electron beam is  
                                                   clampl sample e nP f Nε Σ= ⋅ ⋅ + ,                                                (13) 
where 1,max
sample ph amplNε ω α=  is the average energy in a sample,  is the noise power. This is the 
maximal limit for the power corresponding to the case if all electrons are involved in the 
cooling process simultaneously (screening is absent and the amplification time interval of the 
amplifier  is higher then the time duration of the electron bunch 
amplt∆ bt∆ ).  
   The initial phases inϕ  of electrons in their URWs radiated in the pickup undulator and 
transferred to the entrance of the kicker undulator(s) depend on their energies and amplitudes 
of betatron oscillations. If we assume that synchronous electron enter the kicker undulator 
together with their URW at the decelerating phase corresponding to the maximum 
decelerating effect, then the initial phases for other electrons in their URWs will correspond 
to deceleration as well, if the difference of their closed orbit lengths between undulators 
remains 1,min / 2s λ∆ < . In this case the amplitudes of betatron oscillations, the transverse 
horizontal emittance of the beam in the smooth approximation and the energy spread of the 
beam at zero amplitude of betatron oscillations of electrons must not exceed the values  
,lim 1min , /x x x betA A λ λ π<< = ,     1min2xε λ< ,       
p k c lE E L
σ σ λ β
< = ,           (14) 
where ,p kL  is the distance between pickup and kicker undulators along the synchronous orbit, 
, ,ln / lnc l p kd T d pη = −  is the local slippage factor between the undulators,  is the momentum 
of an electron,  is the pass by time between pickup and kicker undulators. In 
accordance with the betatron phase advance 
βcLT kpkp /,, =
,2 ( 1/ 2)
x p kkψ π= + ; the value ,p kL =  
, ,( 1/x bet p kk 2)λ + , where , /x bet xC vλ =  is the wavelength of betatron oscillations, C is the 
circumference of the ring, xv  is the betatron tune.  
     The third equation in (14) can be overcome if the isochronous bend or bypass between 
undulators will be used. In some cases controllable variable in time optical delay-line can be 
used to change in situ the length of the light pass-way between the undulators during the 
cooling cycle to keep the decelerating phases of electrons in the kicker undulator in the 
process of cooling [6], [7].  
   Below we investigate this case in more details. The difference  in the propagation time 
of the URW and the traveling time  of the electron between pickup and kicker undulators 
depends on initial conditions of electron’s trajectory which can be expressed as   
,p kT
1 0 0 2
0 0 0 0
s s s s
s s s s
x C S E
c t c d c x d x d d
τ τ τ
ρ ρ ρ β
′∆ = − = − − −∫ ∫ ∫ ∫ τρ , 
where , ηβ xxx += ηβ 000 xxx += ,  is the deviation of the electron from its closed orbit, 
 is the deviation of the closed orbit itself from synchronous one,  and  stand for 
appropriate deviations at location 
ηx β0x η0x
0s s= . Two eigenvectors called sine-like S(z) and cosine-
like C(z) trajectories and ρ  stands for the local bending radius,  is a constant which is 
determined by the optical delay line. Basically vectors S(z), C(z) describe the trajectories with 
initial conditions like 
0 0 0 0 0( ) 0; ( , ) ( , )x s x s s x C s′ = = ⋅ s  and 0 0 0 0 0( ) 0; ( , ) ( , )x s x s s x S s′ s= = ⋅ , where s0 
corresponds to the longitudinal position of the pick up. So the transverse position of the 
particle has the form [12] 
)/(),(),(),()( 200000 EEssDssSxssCxsx β∆⋅+⋅′+⋅=  
where   is the deviation of the electron energy from the dedicated energy  and 
dispersion D defined as  
dE E E∆ = − dE
ssSssD
∫∫ ⋅−⋅−=
00 )(
),(),( ,  
Dispersion D(s,s0) describes transverse position of the test particle having relative momentum 
deviation from equilibrium as big as /p p∆ , while its initial values of transverse coordinates 
at s=s0 are zero. So full expression for transverse position of particle comes to form   
0 0 0 0 ,0 0 ,0 0 0 2( ) ( , ) ( , ) [ ( , ) ( , ) ( , ) ]x x
x s x C s s x S s s C s s S s s D s s
′ ′= ⋅ + ⋅ + ⋅ + ⋅ + , 
where xη describes periodic solution for dispersion in damping ring (slippage factor) and 
 marks pure betatron part in transverse coordinate; ββ 00 , xx ′ 0, 0,,x xη η′  stand for its value at 
location of pickup kicker. So the time difference becomes  
51 0 0 52 0 0 56 0 , ,2( , ) ( , ) ( , )t t p k c l
c t c R s s x R s s x R s s c cT
′∆ = − ⋅ − ⋅ − ≅ + ,                 (15) 
where we neglected terms responsible for the betatron oscillations (i.e. R51=0, R52=0).  
     In general case , 0c lη ≠  the initial phase of an electron in the field of amplified URW 
propagating through kicker undulator, according to (15), 1,max 0in tϕ ω= ∆ ≠  and the rate of the 
energy loss   
max| | sin( ) (loss loss in )P P fϕ= − ⋅ ∆E ,                                          (16) 
where ( ) 1 | ( ) | / 2inf E E Mϕ π∆ = − ∆ , if  | | 2in Mϕ π≤  and ( )f E 0∆ =  if | |inϕ > 2 Mπ . The 
function  takes into account that electron with some energy  and its URW enter 
kicker undulator simultaneously at the phase 
(f E∆ ) dE
| | 0inϕ =  and passing together all undulator 
length zero rate of the energy loss if . Electrons having the energies  so they 
and their URWs enter the kicker undulator non-simultaneously with different phases, travel 
together shorter distance in the undulator under smaller rate of the energy change.  
0tc = dE E≠
     According to (16), electrons with different initial phases are accelerated or decelerated and 
gathered at phases 2min mϕ π π= +  ( M m M− ≤ ≤ , 0, 1,..m M= ± ± ) and at energies  
                           
1,min
1,max , , , ,
(2 1)(2 1)
2m d d dp k c l p k c l
E E E E
λ βπβ
ω η η
= + = + ,                       (17) 
if RF accelerating system is switched off (see Fig.2).  
Figure 2: In the  EOC scheme electrons are grouping near the phases 2in mϕ π π= +  (energies mE ) 
     The energy gaps between equilibrium energy positions have magnitudes given by  
                                           1,min 2
gap m m d
p k c l
E E E E
= − = β .                                           (18) 
     Note that the energy gap (18) is 2 times higher the limiting energy spread of the beam at 
zero amplitude of betatron oscillations of electrons (14).  
     The power loss  is the oscillatory function of energy |lossP |dE E− with the amplitude 
linearly decreasing from the maximum value  at the energy max| lossP | dE E=  to a zero one at the 
energy | | . If the RF accelerating system is switched off, the electron energy 
falls into the energy range | |
d gapE E M Eδ− ≥ ⋅
dE E− < gapM Eδ⋅ , excitation of synchrotron oscillations by 
non-synchronous photons can be neglected then the electron energy is drifting to the nearest 
energy value mE . The variation of the particle’s energy looks like it produces aperiodic 
motion in one of 2M  potential wells located one by one. The depth of the wells is decreased 
with their number | . If the delay time in the optical line is changed, the energies |m mE  and 
the energies of particles in the well are changed as well. In this case particles stay in their 
wells if their maximal power loss satisfies the condition    
                                             ,                                                   (19) max| | | / extloss m lossP dE dt P> + |
where | |  stands for the external power losses determined by synchrotron radiation.  extlossP
3. VARIANTS OF OPTICAL COOLING 
Depending on the local slippage factor and coefficients R51, R52 and R56 in (14), different 
variants of optical cooling can be suggested.  
     1. The local slippage factor 0, =lcη , betatron oscillations are absent, dispersion function 
in the pickup undulator , 0x pickupη ≠ . In this case  and the initial phase for all electrons 
can be installed 
t constδ =
inϕ = / 2π . It corresponds to electrons arriving kicker undulator in 
decelerating phases of theirs URWs under maximum rate of energy loss. In this case 
electrons will be gathered near to the synchronous electron if a moving screen opens the way 
only to URWs emitted by electrons with the energy higher than synchronous one. This is the 
case of an EOC in the longitudinal plane based on isochronous bend and screening technique.  
     If electrons develop small betatron oscillations (betatron oscillations introduce phase shift 
less than π/2, (see (14)), then the electron beam will be cooled in transverse and longitudinal 
directions simultaneously. If the dispersion function value in the pickup undulator 0xη =  or if 
the synchrotron oscillations of electrons are small (no selection in longitudinal plane) then the 
cooling in the transverse direction only takes place ( , 0x kickη ≠ ).  
     2. The scheme of OSC can be used at 0, =lcη  [8]. In this scheme the pickup undulator is a 
quadrupole one and kicker undulator is ordinary one. They have the same period. The 
magnetic field in the quadrupole undulator is increased with the radial coordinate   by the 
( )B x G≅ ⋅ x  and changes the sign at 0x = , where G stands for the gradient. The phase of 
the emitted URWs changes its value on π  at 0x =  as well. That is why electrons are 
grouped around synchronous orbit in the ring where they do not emit URWs. The deflection 
parameter in the quadrupole undulator increased with the radial coordinate  and 
so the emitted wavelength also 
( | ( ) |)K B x∝
1min (1 ( )) / 2u K x
2λ λ≅ ⋅ + γ . As the resonance interaction of 
URW and the electron emitted the URW is possible in the kicker undulator only if deflection 
parameters of undulators are near the same, this opens a possibility for initial selection of 
amplitudes in the pickup undulator. So the cooling can be arranged for some specific 
amplitude of synchrotron oscillations only. The continuous resonance interaction and cooling 
is possible if the magnetic field of kicker undulator is decreased in time for cooling process. 
Electrons having other than resonance synchrotron amplitude do not interact with cooling 
system. Betatron oscillations in this scheme must introduce the phase shift less then 2/π  as 
well. This can be arranged by proper zeroing cos and sin-like trajectory integrals [13].  
  The scheme with two quadrupole undulators (the pickup one and the kicker one) described 
in [13]. In this case the second quadrupole undulator decreases the amplitudes of synchrotron 
oscillations for positive deviations  (we choose the conditions when electrons are 
decelerated in their URWs if  and betatron amplitudes are neglected 
( )) and increases them for negative ones 
, 0pickupxη >
, 0pickupxη >
, , 0pickup kickx xβ β= = , 0pickupxη <  as it experience 
deceleration again (the phases of URWs change their value on π  at 0x = and simultaneously 
the electron will pass the kicker undulator at opposite magnetic field). In this case to cool 
electron beam additional selection of URWs can be used by the screen (cut off URWs 
emitted by electrons at negative deviations ).  Another scheme which can be used, based on 
truncated undulator with the magnetic field of the form 
0( ) |xB x G> ⋅ x  and . 
Such undulator can be linearly polarized one with upper or down array of magnetic poles. It 
was used in the undulator radiation experiments in circular accelerators [14].  
0( ) | 0xB x <
     3. If  , 0c lη ≠ , betatron oscillations of electrons introduce phase shift / 2π<<  and the 
energy gaps have the magnitude gapEδ = ,0(3 5) Eσ÷ ⋅ , then transit-time method of OSC based 
on two identical undulators can be used [9]. In this case the main part of electrons including 
tail electrons will be gathered at the energy sE , if the energy 0 |m mE E =0=  was chosen equal 
to sE . Decreasing of the beam dimensions leads to decrease of the rate of cooling. In this 
case a time depended local slippage factor  can be used to decrease the energy gap for 
cooling process and to increase the rate of cooling.  
, ( )c l tη
     4. If  , 0c lη ≠ , the energy gaps between equilibrium energy positions have the magnitudes  
, RF accelerating system is switched off  then electrons are gathered 
at phases 
.0(3 5) /gap EEδ σ≅ ÷ ⋅ M
inϕ  and energies  independently on amplitudes of betatron oscillations. If the 
screen overlap URWs emitted by electrons at negative deviation from one having minimum 
energy  and optical system change the delay time of the rest URWs to move the energies 
min ,0(3 5) EE E σ> + ÷ ⋅  to the energy  then electrons loose their energy and amplitudes of 
betatron oscillations until their energy takes minimum one. Cooling takes place according to 
the scheme of the EOC.  
     5. For this variant  , 0c lη ≠ , the energy gaps between equilibrium energy positions have the 
magnitudes  , the RF accelerating system of the storage ring is switched 
off, the screen absorbs the URWs emitted by electrons at a negative deviation of theirs 
position from the synchronous one in the radial direction, energy layers are located at positive 
deviations from synchronous one outside the energy spread of the beam and optical system 
change the delay time of the URWs to move the energy layers to the synchronous energy. 
Then the energy layers capture small part of electrons of the beam first and electrons with 
smaller energy are captured increasingly and loose their energy and betatron amplitudes until 
reaching the minimum energy  allowed in the beam. So the cooling process takes place. 
This process can be repeated. In this case the energy jump  of the electron 
in the kicker undulator must be less than the energy gap 
,0 /gap EEδ σ<< M
max max| | /loss loss phE P f Nδ =
gapEδ  determined by synchronous 
photons (18) and non-synchronous photons in the URWs having higher energy jumps 
 maxnon synEδ − = max ,loss phE Nδ Σ  at . That is why the next condition must be fulfilled  , 1phN Σ >
                                  
1,minmax 2
phloss ph N gap d
p k c l
E N E E
ηΣΣ >
< = β
.                                            (20)  
Otherwise electrons can jump over the energy layers and cooling will not be effective.  
     If the RF accelerating system of the storage ring is switched on, the average energy loss 
per turn  is higher than the energy loss max| | /turnloss lossE P∆ = 0| (sin sin )seV φ φ−  of the electron in 
the RF accelerating system of the storage ring or 
                                     ,                                           (21) 0| sin sin | /
s lossE eVφ φ δ− <
then the energy of electrons is drifting to the nearest energy value  and EOC takes place. 
Here  is the amplitude of the RF accelerating voltage,  is the synchronous phase 
determined by the equation ,  is the energy 
loss per turn, 
0V sφ
, 0 0sin / /s SR s sE eV V Vφ = ∆ = , , /SR s SR s sE P f eV∆ = =
2 2 22
, 3SR s eP cr B γ=  
3 42.77 10 / s sR Rγ⋅  eV/sec is the average power of the synchrotron 
radiation emitted by the electron in the ring. The value  eV/turn.  7 4, / 5.8 10 /SR s sP f Rγ
     To keep the condition (21) satisfied, the range of RF phases of particles  2 |  
interacting with their URWs must be limited by the value  determined by equality 
in (21). This can be done by using OPA with short amplification time interval 
|sφ φ−
,12 | |c sφ φ−
,1amplt∆  
corresponding to the range of phases   and by overlapping the center 
of this time interval with synchronous particle, where 
RF ampltω ⋅ ∆ < ,12 | |c sφ φ−
2RF RFfω π= , RFf  is the frequency of 
the RF accelerating system of the ring. The last condition is equivalent to  
                            ,                 (22) ,1 ,12 | | /
laser
ampl ampl c s RFl l c φ φ ω< = −
where  is the length of the amplified laser bunch. In this case 2  electron ellipses 
appear around synchronous phase in the longitudinal plane. The amplitudes of synchrotron 
oscillations are determined by the energies  which are moved to synchronous energy if the 
optical system changes the delay time of the URWs. The condition (20) must be fulfilled if 
the RF accelerating system of the storage ring is switched on as well.  
,1ampll 1M +
     The electrons will be gathered effectively on elliptic trajectories having maximum 
energies  (17) if conditions (20), (21) are fulfilled, and the deviation  of 
the electron energies from  in the process of interaction are small. The last condition can 
be overcame by limiting the amplification time by interval 
mE mE E Eδ = −
,2am plt∆  and the corresponding 
length  to  ,2 ,2ampl ampll c t= ∆
                                                ,0,2
,02 2
s glaser laser
ampl ampl
< = ap ,                                                (23) 
where  is the initial length of the electron bunch. Above we suggested that electrons are 
moving along elliptical trajectories 
2 21 /EE sσ∆ = ± − sσ  and interact with URWs at the top 
energies  in the region of the energy deviations .  mE∼ / 8m gapE E E Eδ δ= − =
     Multiple processes of excitation of synchrotron oscillations by non-synchronous and noise 
photons will increase the widths of the electron ellipses and to transfer electrons from one 
ellipse to another. They can be neglected if the equilibrium energy spread (12) of the beam 
is less then the energy gap (18)or  
                                                            ,
E O C
E e q g a pEσ δ<                                                (24) 
     Damping time (10) in variant 5 will be increased  times:   ,02 /s amplσ l
                           ,0 ,0max
E sEOC
loss amplP l
τ =                    
,0 ,0
s x sEOC
loss x kick ampl
β σ σ
= .                                    (25) 
     The variant 5 permits to avoid any changes in the existing lattice of the ring (isochronous 
bend, bypass). It works easier for existing ion storage rings (see Appendix 2).  
     The screen permits us to select in pickup undulator electrons with positive deviations of 
both betatron and synchrotron oscillations, and such a way to produce effective cooling both 
in the transverse and longitudinal direction (we suggested 0xη ≠  in pickup and kicker 
undulators in this case). Using the number of kicker undulators  permits to cool the 
beam either in two directions or in the transverse or in the longitudinal directions only by 
selecting corresponding distances between kicker undulators [4], [6].  
1kickN >
4. OPTICAL SYSTEM FOR THE EOC SCHEME  
UR of an electron gets its well known properties only after the electron passed the undulator 
and UR is considered in far zone. The lens located near the pickup undulator can strongly 
influence to the UR properties if its focus is inside the undulator [15]. 
    For effective cooling of an electron beam in a storage ring, parameters of the beam under 
cooling and the optics of EOC system must fulfill certain requirements.  
   1. The URW, emitted in a pickup undulator must be filtered and passed through the laser 
amplifier.  
   2. In variant 1 each electron in a beam should enter kicker undulator simultaneously with its 
amplified URW emitted in a pickup undulator and to move in decelerating phase of this 
URW. For the test electron of a beam (for example, for the synchronous electron with zero 
amplitude of betatron oscillations) this requirement is satisfied by equating the propagation 
time of the URW with the traveling time of the electron between undulators. Conditions (14) 
are necessary for other electrons of the beam to get decelerating phases of theirs URWs in 
this case.  
    3. Each electron in the beam should enter the kicker undulator with its URW emitted in the 
pickup undulator near the center of this wavelet in transverse direction. This requirement will 
be satisfied if the transverse sizes of all URWs in the kicker undulator are overlapped.     
   The rms transverse size of one URW at the distance l  from the pickup undulator is equal to 
 (assuming that radiation is emitted from the center of the undulator). 
At the distance l  from the undulator the R.M.S. transverse size of the beam of emitted URWs 
is equal , where d  is the transverse size of the electron beam. 
If the optical screen opens the way to radiation from the part of the electron beam only, so the 
only small angles to the undulator axis   passed through, then at the distance from 
the end of the undulator , where   
( ) ( /2)w c ul Mσ θ λ∆ +
, ( ) /2 ( )w b c u cd Mσ θ λ lθ∆ + ∆ ⋅+
( )Cϑ ϑ∆ < ∆
cl l=
                         
( ) 2
                                                       (26) 
URWs will be overlapped, and the transverse size of the beam of the selected wave packets 
will be equal 2 (  (see. Fig. 3). If the beam of URWs passed optical lenses, 
movable screen, the optical amplifier, optical delay and injected into the kicker undulator 
then electrons under cooling will hit theirs URWs in the transverse plane if the transverce 
dimensions of the electron beam in the undulator are less then .  
)cd Mθ λ+ ∆ u
,2 w bσ
  4. Generally the angular resolution of an electron bunch by an optical system is   
                              1min1.22res D
δ ϕ ≅ ,                                                          (27) 
where  is diameter of the first lens. This formula is valid, if elements of the source emit 
radiation which is distributed uniformly in a large solid angle. In our case only a fraction of 
the lens affected by radiation, as . That is why the effective diameter  must 
be used in (27). At the distances   the size ,  so the space resolution of the 
optical system is 
,w bD σ> ,eff w bD σ=
cll > , ( )w b c lσ ∆θ
res resx lδ δ ϕ≅  or  
1min 1min1.22 /( ) 0.86res c ux Lδ λ θ λ≅ ∆ = ,                                     (28) 
where 2( ) ( ) / (1/ ) (1 ) /c c K Mθ ϑ γ γ∆ = ∆ = +  is the observation angle.  
     Note that at closer distances , the spatial resolution is better. More complicated 
optical system can be used for increase the spatial resolution in this case.  
cl l<
                                                       Figure 3: Scheme of URW’s propagation.  
     5. URWs must be focused on the crystal of OPA. For the URW beam, the dedicated 
optical system with focusing lenses can be used to make the Rayleigh length equal to the 
length of the crystal (typically ~1 cm) for small diameters of the focused URW beam in the 
crystal (typically ~0.1 mm).  
     6. The electron bunch spacing in storage rings is much bigger than the bunch length. The 
same time structure of the OPA must take advantage on this circumstance.   
5. USEFUL EXPRESSIONS FROM THE THEORY OF CICLIC ACCELERATORS 
The equilibrium value of relative energy spread of the electron beam in the isomagnetic 
lattice of the storage ring determined by synchrotron radiation (SR) described by expression  
655 6.2 10
eq SR
s s sE R R
γ γ−Λ= ⋅
,                                       (29) 
where  cm, 11/ 3.86 10c mc
−Λ = ⋅
sR  is the equilibrium bending radius of the synchronous 
electron, in the smooth approximation 4 /s c s sR Rαℑ = −  is a coefficient determined by the 
magnetic lattice ( ),  is the momentum compaction factor, 0 3s< ℑ < cα sR  is the equilibrium 
averaged radius of the storage ring [16].  
     For small synchrotron oscillations the equilibrium length of the electron bunch is 
eq SR
eq SR c E
,                                                           (30)  
where 0 cos /2rev c s sh eV Eω α φ πΩ=  is the synchrotron frequency, 2rev fω π= , h  is an 
accelerating RF voltage harmonic order.  
     The equilibrium radial synchrotron and betatron beam dimensions are  
eq SR
eq SR E
x x Eη
σ η= ,           , 55 3 6.2 10
eq SR s
σ γ γ−
= = ⋅
,                       (31) 
where in the smooth approximation /s c s sR Rα 1ℑ = −  is a coefficient determined by the 
magnetic lattice ( ),  is a coefficient [16],  [17].  3x sℑ +ℑ = 1F ∼
      The damping time in the storage ring determined be synchrotron radiation in the bending 
magnets and comes to the following values ( 1zℑ = )  
                            , , 3 3
, , , , , , ,
355SR s sx z s
SR s x z s e x z s x z s
s sR R RE
= = =
ℑ ℑ ℑ
[sec].                                          (32) 
     The maximum deviation of the energy from its synchronous one is  
                            0
[( 2 )sin 2 cos ]sep s s
E h E
β π φ φ
= ± − − sφ ,                                      (33) 
where  is the storage ring slippage factor of the ring.  2/1 γαη −= cc
     For the ordinary storage ring lattice (without local isochronous bend or bypass between 
undulators) the natural local slippage factor 
                                                                  
p kc l c
L Cη η .                                                      (34) 
6. EXAMPLE 
Below we consider an example of one dimensional EOC of an electron beam in the 
transverse x-plane in strong focusing storage ring like Siberia-2 (Kurchatov Institute Atomic 
Energy, Moscow) having maximal energy 2.5 GeV [18]. The magnetic system of the ring is 
designed with so-called separate functions. The lattice consists of six mirror-symmetrical 
superperiods, each containing an achromatic bend and two 3 m long straight sections. For the 
functionality, the half of the superperiod arranged with two sections. The first one, 
comprising the quadrupoles F,D,F and two bending magnets is responsible for the achromatic 
bend and high xβ , yβ  functions in the undulator straight section. The second part, 
comprising quadrupoles D,F,D and dispersion free straight section, allows to change the 
betatron tunes, without disturbing the achromatic bend. Main parameters of the ring, the 
electron beam in the ring, pickup and kicker undulators and Optical Parametric Amplifier are 
presented in Tables 1 - 4.  
      After single bunch injection in the storage ring the energy 100 MeV is establishd for the 
experiment and the beam is cooled by synchrotron radiation damping (see section 5). In this 
case the energy spread and the beam size acquire equilibrium values in ~40 seconds (see 
Table1). The equilibrium energy spread is equal to , / 3.94 10eq SRE Eσ 5−= ⋅ , the length of the bunch 
 cm at the amplitude of the accelerating voltage  =  73 V, the synchronous 
voltage  V, the radial emittance  cm
, 2.32eq SRsσ = 0V
1.89sV =
, 6 21.25 10 [ ]eq SRx E GeV
−∈ = ⋅ 81.25 10−= ⋅  rad, the radial 
betatron beam dimension at pickup undulator , 24.61 10eq SRxσ −= ⋅  mm.  
     Following synchrotron radiation damping the amplitudes of radial betatron oscillations 
,0xσ  are artificially excited to be suitable for resolution of the electron beam in the experiment 
with EOC (see Table 2). The amplitudes of  synchrotron oscillations must stay damped to 
work with short electron bunches and short duration of amplification OPAs.  
     In the variants of the example considered below the optical system resolution of electron 
beam, according to (28), is 1.9resxδ =  mm at 
1,min 2 10λ
−= ⋅  cm, 240uMλ =  cm. It yields that 
effective EOC in this case is possible if the beam under cooling has total size in the pickup 
undulator , 2.0x totσ >  mm. We accepted the initial energy spread , the 
dispersion beam size  mm, the length of the electron bunch   
cm, its transverse size at pickup undulator 
,0 3.94 10
eq SR
E E Eσ σ
−= = ⋅
,0 3.15 10xησ
−= ⋅ ,0sσ =
,2 4eq SRsσ = .64
,0 4xσ =  mm,  the laser amplification length 
 mm (duration 0.5 ps), the radial betatron beam size in kicker undulator 1.5laserampll = ,0 1xσ = mm, 
the URW beam size  mm. We took the number of electrons at the orbit . In 
this case the number of electrons in the URW sample is 
, 2w bσ =
, 5 10eN Σ = ⋅
, 129.5e sN = , the number of non-
synchronous photons in the sample is , 2.5phN Σ =  for the case of one noice photon at the OPA 
front end. In this storage ring the natural local slippage factor (34) is , , /c l c p kL Cη η=  
, /c p kL Cα =
34.45 10−⋅ , the energy gap (18) is 0.62gapEδ =  keV. 
     We consider EOC in the transverse plane. In this case the dispersion beam size 
,0x resxησ δ<<  and that is why there is no selection of electrons in the longitudinal plane. That is 
why in order to prevent heating in the longitudinal plane by energy jumps determined by both 
synchronous and non-synchronous photons in the URWs, two kicker undulators are used 
which produce zero total energy jump [4], [6]. Note that the purpose of this experiment is to 
check physics of optical cooling. At the same time cooling in the transverce plane is 
important for heavy ions in RHIC, LHC.  
     We accept the distance between pickup and first kicker undulator along the synchronous 
orbit  m (, 72.27p kL = ,9 , 4
x p kkψ π= = ). It corresponds to the installation of undulators in 
the first and seventh straight sections which are located at a distance 72.38 m (we count off 
pickup undulator). Second kicker undulator is located on the same distance from the first one. 
Optical line is tuned such a way that electrons are decelerated in the first kicker undulator and 
accelerated in the second one.  
     The URWs have the number of the photons emitted in the pickup undulator (see Table 3) 
 per electron in the frequency and angular ranges (3) suitable for cooling. The 
limiting amplitude of betatron oscillations (14) is 
21.15 10phN
,lim 3.2xA =  mm. The energy spread of the 
beam limited by the separatrix is 4/ 3.3 10sepE E
−∆ = ⋅ . The electric field strength at the first 
harmonic of the amplified URW in the kicker undulator is clwE ≅ 32.06 10−⋅  V/cm. The power 
loss for the electron passing through one kicker undulator together with its amplified URW 
comes to  eV/sec if the amplification gain of OPA is (see Tables 2, 
4). This power loss corresponds to the maximal energy jumps  eV and the average 
energy loss per turn  eV/turn. The jump of the closed orbits is 
max 62.03 10lossP = ⋅
710amplα =
max 73lossE∆ =
0.84turnlossE∆ =
1 55.8 10xηδ
−= ⋅  cm. 
Below we will consider two variants of EOC. 
     1. The variant 1 (section 3). For the parameters presented above the cooling time for the 
transverse coordinate, according to (10), comes to , 18.5x EOCτ =  msec. SR damping time ~ 40 
sec is much bigger (see Table 1). The average power transferred from the optical amplifier to 
electron beam (13) is 0.061amplP =  mW. It is determined by the power of the URWs (0.036 
mW) and noise average power (41) equal to 0.025 mW. We adopted one-photon/mode (one-
photon/sample) at the amplifier front end corresponding to pulse noise power at the amplifier 
front end W,  W at the gain , used  eV. 
We took into account that the amplification time interval of the amplifier  is less than 
the revolution period by a factor of 
,0 /nP c Mω λ= = 74 10−⋅
2nP = 70 10G = 0.5ω
amplt∆
,0/ / 2 8.27b sC c t C σ∆ = = ⋅  times.  
     Necessary conditions for selection of electrons must be created: high beta function in the 
pickup undulator (to increase the transverce dimensions of the bunch for selection of 
electrons with positive deviations from closed orbit), the isochronous bend between 
undulators. We believe that the lattice of the ring is flexible enough to be changed in 
nesessary limits by analogy with those presented in [19].  
    The number of  electrons in the bunch is enough to detect them in the experiment and to 
neglect intrabeam scattering.  
     Note that if one kicker undulator is used in the scheme of two-dimensional EOC and the 
beam resolution is high  mm, the equilibrium relative energy spread, the spread of 
closed orbits, the longitudinal, dispersion and radial betatron beam dimensions determined by 
EOC, according to (11), are equal to 
210resxδ
, 5/ 5.56 10eq EOCE sEσ
−= ⋅ ,,0 2.63
eq EOC
sσ =,  cm,  , ,eq EOC eq EOCx xησ σ= =
24.46 10−⋅  mm. It follows that if the number of electrons in the bunch  then their 
influence on the equilibrium dimensions of the bunch can be neglected, longitusinal 
dimensions and the energy spread stay small and the radial betatron beam dimensions 
determined by EOC are high degree decreased. In reality the equilibrium synchrotron and 
betatron bunch dimensions will be much higher. This is the consequence of the finite beam 
resolution in pickup undulator. That is why we use two kicker undulators to keep longitudinal 
bunch dimensions small to exclude the excitation of longitudinal oscillations by multiple 
energy jumps. The situation could be better if we had effective OPA at the wavelength 
 cm, i.e. about one order less. Shorter undulator can be used as well.  
, 5 10eN Σ < ⋅
1,min 3 10λ
      2. The variant 5 (section 3).  The variant 5 requires easier tuning of the lattice for the 
arrangement of the local small slippage factor between undulators. In the case of one-
dimensional EOC, using two kicker undulators, the multiple processes of excitation are not 
essential because of the excitation of the synchrotron oscillations in this case is absent or 
unessential and that is why there is no need in the small local slippage factor. In this case the 
initial phase ( , )in xE Aϕ  of the electron in the field of amplified URW propagating through the 
kicker undulator, according to (15) is the function of both the energy (which is a constant in 
this variant of EOC) and the amplitude of betatron oscillations. The amplitudes of betatron 
oscillations will increase or decrease depending on their initial phases until they reach the 
equilibrium amplitudes determined, in the smooth approximation, by the expression 
, 1min ,2x m x betA m /λ λ= ⋅ π  (generelised expression (14)) corresponding to the phases 2m mϕ π= . 
Variable in time optical delay-line can be used to change in situ the length of the light pass-
way between the undulators during the cooling cycle to move the initial phases to 
2in m / 2ϕ π π+  for production of the optimal rate of decrease the amplitudes of betatron 
oscillations of electrons in the fields of amplified URW and the kicker undulator. The 
damping time for radial betatron oscillations, according to (25), is , 0.57x EOCτ =  sec.  
     Note that in the case of two-dimensional EOC using one kicker undulator, according to 
(18), the energy gaps between equilibrium energy positions have the magnitudes 
0.62gapEδ =  keV. They are higher than the energy jumps of electrons in kicker undulator 
 eV and the energy jumps max 73lossE∆ = max , 115loss phE N Σ∆ =  eV determined by the non-
synchronous photons in the URWs (see condition (20)). The conditions (22), (23) or 
 limit the length of the laser URWs by the values   cm,  cm. 
The accepted laser amplification length  mm is enough to satisfy these conditions. 
The damping time for radial betatron oscillations, according to (25), is 
laser
ampl ampll l< ,1 1.69ampll = ,2 0.32ampll =
1.5laserampll =
, 1.14x EOCτ =  sec. This 
damping time is less then one for synchrotron radiation damping (see Table1). The 
equilibrium energy spread determined by EOC is about 6-10 times higher then the energy 
gap. It follows that the local slippage factor between undulators, according to (24), must be 
decreased by a factor higher then 10. Unfortunately the resolution of the electron beam will 
not permit to reach the equilibrium energy spread and cooling in the transverse plane in this 
case will be less then heating in the longitudinal one.  
7. CONCLUSIONS 
  We have shown in this paper, that test of EOC is possible in the 2.5 GeV electron strong 
focusing storage ring tuned down to the energy ~ 100 MeV. Electron beam can be cooled in 
transverse direction. The damping time is much less than one determined by synchrotron 
radiation. So the EOC can be identified by the change of the damping rate of the electron 
beam. Variant of cooling is found, which permits to avoid any changes in the existing lattice 
of the ring (for production of isochronous bend, bypass). It can work for existing ion storage 
rings as well (see Appendix 2). Three short undulators in this variant installed in the storage 
ring have rather long periods and weak fields. They can be manufactured at low cost.  
   The cooling of a relatively small number of electrons (one bunch, ) is considered 
in this proposal in attempt to avoid strong influence of the non-synchronous photons on the 
equilibrium energy spread of the beam. The intrabeam scattering effects could be overcame 
as well. Optical amplifier suitable for the EOC - so called Optical Parametric Amplifier - 
suggested as a baseline of experiment, must have moderate gain and power. We have chosen 
the wavelength of the OPA equal 2 mkm as the OPA technique is more developed for these 
wavelengths. At the present times the OPAs having amplification gain ~10
, 5 10eN Σ = ⋅
8 and the power >1 
W are fully satisfy requirements for this experiment (See Appendix 3). Usage of OPAs with 
shorter wavelength will permit to increase the spatial resolution by the optical system and the 
degree of cooling of the beam.   
     We have predicted that the maximum rate of energy loss for electrons in the fields of the 
kicker undulator and amplified URW calculated in the framework of classical 
electrodynamics is 9.3phN  times lesser then one taking into account quantum nature of 
the photon emission in undulators. Quantum aspects of the beam physics will be checked in 
the proposed test experiment. It is suggested that the scheme based on optical line with 
variable delay time will be tested as well.  
     Authors thank A.V.Vinogradov and Yu.Ja.Maslova for useful discussions.  
     Supported by RFBR under grant No 05-02-17162a, 05-02-17448a, by the Program  
     of Fundamental Research of RAS, subprogram “Laser systems” and by NSF.   
Table 1.  Parameters of the ring:  
The maximal energy of the storage ring                                      Emax= 2.5 GeV 
The energy for the experiment                                                     Eexp=100 MeV 
Relativistic factor for the experiment                                          200γ ≅  
Circumference                                                                              C=124.13 m 
Bending radius                                                                              cm  490.54R =
Average radius                                                                             1976R=  cm 
Frequency of revolution                                                              62.42 10f = ⋅  Hz 
Harmonic number                                                                        75h =  
RF frequency                                                                               181.14RFf =  MHz 
Energy loss determined by SR                                                    , / 1.8SRP fγ 9=  eV/turn 
The amplitude of the accelerating voltage at                         =  73 V  expE 0V
The synchronous phase                                                                 0.026sϕ
Radial tune                                                                                   7.731xv =  
Vertical tune                                                                                 7.745zv =  
Momentum compaction factor                                                      37.6 10cα −= ⋅
Dispersion function at the pickup and the kicker undulator        80xη =  cm,  
Radial beta function in pickup undulator                                     17xβ =  m  
Radial beta function in kicker undulator                                     1.7xβ =  m 
Vertical beta function                                                                   6zβ =  m  
Patrician coefficients                 , ,z x sℑ ℑ ℑ                                   1,  0.97, 2.03 
Damping times by SR at                                        43.1; 44.4; 21.23 sec expE , ,z xτ τ τs
The length of the period of betatron oscillations                          , 16.06x betλ = m 
Slippage factor of the ring                                                            c cη α=  
Local slippage factor of the ring                                                   , 0.58c l cη α= ⋅  
Frequency of synchrotron oscillations at 100 MeV           expE = 31.6 10 f−Ω = ⋅ .  
Table 2.    Initial parameters of the electron beam in the ring:  
Number of electrons at the orbit                                           4, 5 10eN Σ = ⋅ ,  
Number of electron bunches being cooled                             1 
Relative energy spread                                                          ,  5,0 / 3.94 10E Eσ
Betatron beam size at pickup undulator ( 32xβ =  m)             ,0 4xσ =  mm,  
Betatron beam size at kicker undulator ( 2xβ =  m)               ,0 1xσ =  mm,  
Dispersion beam size                                                             2,0 3.15 10xησ
−= ⋅  mm,  
Total beam size at pickup undulator                                     2 2, ,0 ,0( ) ( )x tot x xησ σ σ 4= + =  mm,  
The length of the electron bunch                                            cm,  ,0 2.32sσ =
Table 3.  Parameters of pickup and kicker undulators:  
Magnetic field strength                                                              2 1 3 3 8B = Gs,  
Undulator period                                                                        8uλ =  cm,  
Number of periods                                                                      M = 30,   
Deflection parameter                                                                  K=1.  
Table 4.   Optical Parametric Amplifier  
Number of Optical Parametric Amplifiers                                             2 
Total gain                                                                                                 710amplα =
The wavelength of amplified URWs                                                     cm 41,min 2 10λ
The characreristic URW waist size                                                         mm   . 0.77w cσ =
The URW beam waist size                                                                      mm   2wσ =
The duration of the amplification time of the OPA                                5 psec (  mm) 1.5ampll =
The frequency of the amplified cycles                                                     Hz 62.42 10amplf f= = ⋅
Appendix 1  
Spectral-angular distribution of the UR energy emitted by the relativistic electron in the 
pickup undulator on the harmonic  is  n
                                         
sin ( , )
E M E
c σ ω ϑ
∂ ∂ ∂
,                                           (35) 
where  is the angular distribution of the energy of the UR emitted in the unit solid 
angle  at the angle 
/nE∂ ∂o
do θ  to it’s axis, , , sin sin /n n nσc σ σ= ( )/n n nnMσ π ω ω ω= −
1n nω ω=  is the frequency of the -th harmonic of  UR. [20] - [22]. For the helical undulator  n
                                   
2 2 3 2 2 2
( ) ( ) 6 ( )
n n totE e Mn F K E n F K
c K 2 31 )
nω ϑ β ϑ γ ϑ
⊥∂ ,= =
∂ Ω +
n nF K J n
2( ) ( )ϑ χ
2 2 2 21
nK J n
ϑ( ) )χ
+ − ,+  2 2 24 3totE e M K cπ γ= Ω / , 2 ucπ λΩ = / ,  is the 
Bessel function and it’s derivative, 
n nJ J
2 22 (1 ) 1K Kχ ϑ ϑ= / + + < /Kβ γ⊥ =, . The number of 
photons emitted in the undulator on the harmonic  in the solid angle  and 
frequency band dω is determined by the relation  
n 2 /do dπ ϑ γ= 2
                            
2 2 2
, 2 2 2
sin ( , )
ph n n
n M K F K
α λ ϑ
d doσ ω ϑ ω
+ +∫ ∫ .                  (36) 
     If the considered frequency band  then we can neglect the angular 
dependence of the first multiplier and the frequency dependence of the value in (36). 
In this case the value  determine the range of angles of the UR (3). The increasing of 
the frequency band will lead to the increase of the angular range. Taking the frequency band 
 out of the integral and taking into account that 
/ 1/Mω ω∆ 1
sin nc σ
sin nc σ
ω∆ 2 2 1(2 / ) nd Mϑ γ π ω σ= Ω ,  
we can transform (36) to (6) for  sin ( )nc σ πδ σ= n / 1/2Mω ω∆ = 0ϑ =  and .  1n =
Appendix 2  
Below we investigate the possibility of lead ions cooling (Z=82) in the storage ring LHC 
based on using the version 5 (section 3) of EOC. We take the example 2 considered in [7]. In 
this case the energy  eV, (141.85 10E = ⋅ 953γ = ) the slippage factor of the ring 43.23 10c cη α −= ⋅ , 
the amplitude of the RF accelerating voltage  MV, the RMS bunch length 8 cm, RF 
frequency  MHz, synchrotron frequency  Hz, circumference C=2665888.3 
cm, harmonic order h=35640, RMS relative energy spread ,  eV, 
0 16V =
400RFf = 23Ω=
,0 / 1.1 10E Eσ
−= ⋅ 102.04 10inEσ = ⋅
, 415x betλ =  m, RF bucket half-height 
4/ 4.43 10sepE E
−∆ = ⋅ ,  eV. We take the 
distance between pickup and kicker  undulators 
108.19 10sepE∆ = ⋅
, 1453p kL =  m (k=3), synchrotron radiation 
energy loss per ion per turn  eV.  4, / 1.2 10SR sP f = ⋅
     For the parameters of the undulators [7] the energy loss per turn is  eV, 
 eV (M=12), the gap between equilibrium energy positions is 
 eV,  cm. It follows that , that is the condition  (20)  
is satisfied. In the case of ions the equation (21)  must include  instead of . In this 
example . It follows the laser amplification length  mm. By 
this choice the condition (21) is satisfied as well. To cool the ion beam in the transverse plane 
and to keep the magnetic lattice unchanged one pickup and two kicker undulators must be 
used.  
max 53 10lossE∆ = ⋅
,0 / 1.7 1E Mσ = ⋅ 0
81.97 10gapEδ = ⋅
1,min 5 10λ
−= ⋅ maxloss gapE Eδ∆
0eZ V⋅ 0eV
max 4
0/ 2.2 1lossE eZVδ
−= ⋅ ,2 2.78ampll =
Appendix 3  
The principle of OPG is quite simple: in a suitable nonlinear crystal, a high frequency and 
high intensity beam (the pump beam, at frequency pω ) amplifies a lower frequency, lower 
intensity beam (the signal beam, at frequency sω ); in addition a third beam (the idler beam, 
at frequency iω , with i s pω ω ω< < ) is generated (In the OPG process, signal and idler beams 
play an interchangeable role, we assume that the signal is at higher frequency, i.e., s iω ω> ).. 
In the interaction, energy conservation  
                                                                  p s iω ω ω= +                                    
is satisfied; for the interaction to be efficient, also the momentum conservation (or phase 
matching) condition 
                                                                 p s i= +k k k                                      
where , pk sk , and  are the wave vectors of pump, signal, and idler, respectively, must be 
fulfilled. The signal frequency to be amplified can vary in principle from 
2pω  (the 
so-called degeneracy condition) to pω , and correspondingly the idler varies from 2pω  to 0; 
at degeneracy, signal and idler have the same frequency. In summary, the OPG process 
transfers energy from a high-power, fixed frequency pump beam to a low-power, variable 
frequency signal beam, thereby generating also a third idler beam. 
The signal and idler group velocities sv  and  (GVM – group velocity mismatch) 
determine the phase matching bandwidth for the parametric amplification process. Let us 
assume that perfect phase matching is achieved for a given signal frequency 
sω  (and for the 
corresponding idler frequency i p sω ω ω= − . If the signal frequency increases to sω ω+∆ , by 
energy conservation the idler frequency decreases to iω ω−∆ . The wave vector mismatch 
can then be approximated to the first order as 
                                              
s i gi gs
k ω ω ω
ω ω ν ν
⎛ ⎞∂ ∂
∆ ≅ − ∆ + ∆ = − ∆⎜ ⎟⎜ ⎟∂ ∂ ⎝ ⎠
    The gain bandwidth of an OPA can be estimated using the analytical solution of the 
coupled wave equations in the slowly varying envelope approximation and assuming flat top 
spatial and temporal profiles and no pump depletion. The intensity gain (G ) and phase (ϕ ) 
of the amplified signal beam are given in [23] by  
   ( )
2 sinh
γ ⎛ ⎞= + ⎜ ⎟
sin cosh cos sinh
tan ,
cos cosh sin sinh
B A B A A B
B A B A A B
            (37) 
where 2,A kL= ∆  ( ) ( )2 22 ,B L kLγ= − ∆ and 0gain coefficient 4 2 ,eff p p s i s id I n n n cγ π ε= = λ λ
(phase mismatch ,p s ikL L∆ = = − −k k k  where L is the length of amplifier,  is the 
effective nonlinear coefficient, 
pI  is the pumping intensity. 
       The full width at half maximum (FWHM) phase matching bandwidth can then be 
calculated within the large-gain approximation as 
                                            
( )1 2 1 22 ln 2 1
gs gi
⎛ ⎞∆ ≅ ⎜ ⎟
                                              
(38) 
Large GVM between signal and idler waves dramatically decreases the phase matching 
bandwidth; large gain bandwidth can be expected when the OPA approaches degeneracy 
( s iω ω→ ) in type I phase matching or in the case of group velocity matching between signal 
and idler ( gs giν ν= ). Obviously, in this case Eq. (5) loses validity and the phase mismatch k∆  
must be expanded to the second order, giving  
                                       
( )1 4 1 4
ln 2 1
L k k
∆ ≅ ⎜ ⎟
∂ ∂⎝ ⎠
                                                        
For the case of perfect phase matching ( 0k∆ = , B Lγ= ) and in the large gain approximation 
( 1Lγ ), Eq. (37)(4) simplify to 
                       ( ) (1 04 exp 2 ,s s )I L I Lγ≅           ( ) (0 exp 2 .4
)I L I Lω γ
≅                             (39) 
Note that the ratio of signal and idler intensities is such that an equal number of signal and 
idler photons are generated. Equations (39) allow defining a parametric gain as 
                                                      ( )1 exp 2
G Lγ≅     
growing exponentially with the crystal length L  and with the nonlinear coefficient γ .  
            ( ) ( )( )0 01 1exp 8 2 exp 8 24 4eff p p s i s i eff s s p pG L d I n n n c L d n Iπ ε λ λ π λ ε≅ ≅ n c
      (40) 
for  and in n≈ i sλ λ≈ , 
377 Ohm
= .  
     The noise (amplified self emission) power of the optical amplifier is determined by  the 
expression  
                                                            ,                                                   (41) ,0 0n nP P G=
where  is the noise power at the amplifier front end,  is the gain of the amplifier.  ,0nP 0G
If  the noise power corresponds to one-photon/mode at the amplifier front end then 
 [27], [28], where in our case  is the coherence length, 
,0 1max /nP ω τ= coh 2coh MTτ =
1min /T cλ=
Example: MgO Periodically Poled Lithium Niobate 
     In recent years the development of periodically poled nonlinear materials has enhanced 
the flexibility and performance of OPAs. In the case of well-studied Periodically Poled 
Lithium Niobat (PPLN), one can get access to the material’s highest effective nonlinearity as 
well as retain generous flexibility in phase-matching parameters and nonlinear interaction 
lengths.  
     Operation near the degeneracy wavelength of 2.128 µm reduces thermal-lens effect 
because the signal and the idler wavelengths fall within the highest transparency range of 
Lithium Niobat. For wideband optical parametric amplification, we will use MgO: PPLN 
with a poling period of 31.1 (31.2) µm, which has high damage threshold and high nonlinear 
coefficient 16pm/V [25] (17pm/V [26]). To avoid photorefractive damage, thick (~1-2 mm) 
PPLN crystal was suggested to be kept at elevated temperatures 1500C [24]. 
For the signal wavelength sλ  = 2 µm, 2.1s pn n≈ = , and the gain, according to formula (40) 
comes to  
exp 3
pI lG
GW cm mm
≅ ⎜ ⎟
For G=107,  Ip=1GW/cm2,  l = 5.8 mm. We propose two-stage crystal amplifier (l1 = l2 = 3.5 
mm). OPA amplifies linearly polarized radiation. That is why the circular polarization of the 
URWs (if helical undulator is used) in our case must be transformed into linear polarized one 
before it will be injected in the kicker undulator. Usual quarter wave plate can be used for this 
purpose in simplest case. Planar undulators can be used as well. Reflective optics can be used 
for dispersion-free undulator light propagation.  
REFERENCES  
[1] A.M.Sessler, “Methods of beam cooling”, LBL-38278 UC-427, February 1996; 31st 
Workshop:   “Crystalline Beams and Related Issues”, November 11-21, 1995.  
[2] D.Mohl, A.M.Sessler, “Beam cooling: Principles and Achievements”, NIMA, 532(2004), 
p.1-10.  
[3] D. Mohl, “Stochastic cooling”, CERN Accelerator School Report, CERN No 87-03, 1987, 
pp.453- 533.  
[4]. E.G.Bessonov, “On Violation of the Robinson’s Damping Criterion and Enhanced 
Cooling of Ion, Electron and Muon Beams in Storage Rings”, 
http://arxive.org/abs/physics/0404142 .   
[5] E.G.Bessonov, A.A.Mikhailichenko, “Enhanced Optical Cooling of Particle Beams in 
Storage  Rings”, Published in Proc. 2005 Particle Accelerator Conference, May 16-20, 
2005, Knoxville,  Tennessee, USA, (http://www.sns.gov/PAC05),    
    http://accelconf.web.cern.ch/accelconf/p05/PAPERS/TPAT086.PDF.   
[6] E.G.Bessonov, A.A.Mikhailichenko, A.V.Poseryaev, “Physics of the Enhanced optical 
cooling of  particle beams in storage rings”, http://arxiv.org/abs/physics/0509196.    
[7] E.G.Bessonov, M.V. Gorbunkov, A.A.Mikhailichenko, “Enhanced optical cooling of ion 
beams for LHC”, Proc. 2006 European Particle accelerator Conference, June 26-30 2006.  
Edinburgh, Scotland, 
       http://accelconf.web.cern.ch/accelconf/e06/PAPERS/TUPLS001.PDF;  
      Electronic Journal of Instrumentation – 2006 JINST 1 P08005, http://jinst.sissa.it/,        
      http://ej.iop.org/links/r5pyfrsWE/sl44atI92xGGY2iAav5vpA/jinst6_08_p08005.pdf.  
[8] A.A.Mikhailichenko, M.S.Zolotorev,” Optical Stochastic Cooling”, SLAC-PUB-6272, 
Jul1993, 11pp. Published in Phys.Rev.Lett.71:4146-4149,1993. 
[9] M.S.Zolotorev, A.A.Zholents, “Transit-Time Method of Optical Stochastic Cooling”, 
Phys. Rev. E   v.50, No 4, 1994, p. 3087.  
[10] M.Babzich, I.Ben-Zvi, I.Pavlishin at all., “Optical Stochastic Cooling for RHIC Using 
Optical  Parametric Amplification”, PRSTAB, v.7, 012801 (2004).  
[11] E.G.Bessonov, “Some Aspects of the Theory and Technology of the Conversion 
Systems of Linear Colliders”, Proc. 15th Int. Accelerator Conf. on High Energy 
Accelerators, (Int. J. Mod. Phys Proc. Suppl.2A), V.1, pp. 138-141, 1992, Hamburg, 
Germany.  
[12] K. Steffen, “High Energy Beam optics”, Interscience Pub., 1964. 
[13] A.A. Mikhailichenko, “Optical Stochastic Cooling and Requirements for a Laser 
Amplifier”, CLNS-98-1539, Dec 1997, Cornell U., 14pp.; Int. Conf. on Lasers '97, New 
Orleans, LA, 15-19 Dec. 1997, Proceedings, ed. J.J.Carrol, T.A.Goldman, pp. 890-897.     
[14] D.F.Alferov, Yu.A.Bashmakov, K.A.Belovontsev, E.G.Bessonov, P.A.Cherenkov,    
       “Observation of undulating radiation with the "Pakhra" synchrotron”, Phys. - JETP 
Lett.,  1977, v.26, N7, p.385.  
[15] V.I.Alexeev, E.G.Bessonov, “Experiments on Generation of Long Wavelength Edge 
Radiation  along the Directions nearly Coincident with the Aaxis of a Straight Section of 
the  “Pakhra”synchrotron”, Nucl.  Instr. Meth. B 173 (2001), p. 51-60.  
[16] H.Bruck, “Circular Particle Accelerators”, Press Universities of France, 1966.  
[17] A.A.Kolomensky and  A.N.Lebedev, “Theory of Cyclic Accelerators”, North Holland 
Publ., 1966.  
[18] V.V.Anashin, A.G.Valentinov, V.G. Veshcherevich et al., The dedicated synchrotron 
radiation source SIBERIA-2, NIM A282 (1989), p.369-374.  
[19] H.Hama, S.Takano, G.Isoyama, Control of  bunch length on the UVSOR Storage ring, 
Workshop on Foth Generation Light Sources, Chairmen M. Cornacchia and H. Winick,  
1992, SSRL 92/02, USA, p.208-216.  
[20] D.F.Alferov, Yu.A.Bashmakov, E.G.Bessonov. “To the theory of the undulator 
radiation”. Sov. Phys.-Tech. Phys., 18, (1974),1336.  
http://arxive.org/abs/physics/0404142
http://www.sns.gov/PAC05
http://accelconf.web.cern.ch/accelconf/p05/PAPERS/TPAT086.PDF
http://arxiv.org/abs/physics/0509196
http://accelconf.web.cern.ch/accelconf/p05/PAPERS/TUPLS001.PDF
http://jinst.sissa.it/
http://ej.iop.org/links/r5pyfrsWE/sl44atI92xGGY2iAav5vpA/jinst6_08_p08005.pdf
[21]   D.F.Alferov, Yu.A.Bashmakov, K.A.Belovintsev, E.G.Bessonov, P.A.Cherenkov. "The 
Undulator as a Source of Electromagnetic Radiation”, Particle accelerators, (1979), v.9, 
No 4, p. 223-235.  
[22] E.G.Bessonov, Undulators, Undulator Radiation, Free-Electron Lasers, Proc. Lebedev   
       Phys. Inst., Ser.214, 1993, p.3-119 (Undulator Radiation and Free-Electron Lasers),  
       Chief ed. N.G.Basov, Editor-in-chief P.A.Cherenkov (in Russian).   
[23] J.A. Armstrong, N. Bloembergen, J. Ducuing, P.S. Pershan, Phys. Rev. 127 (1962) 1918. 
[24] C.W. Hoyt, M. Sheik-Bahae, M. Ebrahimzadeh, High-power picosecond optical 
parametric oscillator based on periodically poled lithium niobate, Opt. Lett. Vol. 27, No. 
17, 2002. 
[25] K.W. Chang, A.C. Chiang, T.C. Lin, B.C. Wong, Y.H. Chen, Y.C. Huang Simultaneous 
wavelength conversion and amplitude modulation in a monolithic periodically-poled 
lithium niobate. Opt. Comms, 203 (2002) 163-168. 
[26] L. E. Myers, G. D. Miller, E. C. Eckardt, M. M. Fejer, and R. L. Byer, Opt. Let. 20, 52  
      (1995).  
[27] A.Maitland and M.H.Dunn, Laser physics, Noth-Holland publishing Company, 
Amsterdam – London, 1969.  
[28] I.N.Ross, P.Matousek, M.Towrie, et al., Optics communications 144 (1997), p. 125- 133.  
This paper is published in: http://arxive.org/abs/0704.0870   
http://arxive.org/abs/0704.0870
