ENHANCED RADIATIVE ION COOLING
E.G.Bessonov, Lebedev Physical Institute RAS, Moscow, Russia
Abstract
New methods of enhanced laser cooling of particle beams
in storage rings and Robinson’s damping criterion are discussed.

1

INTRODUCTION

In the method of ordinary three-dimensional radiative cooling of ion beams a laser beam overlaps an ion beam, its
transverse position is motionless, all ions interact with thelaser beam independent of their energy and amplitude ofbetatron oscillations. The difference in rates of momentum
loss of ions having maximum and minimum energies in the
beam is small. That is why the cooling time of the ion beamis high [1] - [3].
When a cooling is produced in a dispersion-free straightsection and the laser beam intensity is constant inside thearea of the laser beam occupied by a being cooled ion
beam, then the damping times of the horizontal vertical andphase oscillations are:τx = τy =

2ε
τ
= ,
1+D
P

(1)

where P is the average power of scattered radiation; ε, ion
energy; D, saturation parameter.
The physics of three-dimensional radiative ion cooling issimilar to synchrotron radiation damping. Friction causes
the appearance of a reaction force on the emitting particle. Liouville’s theorem does not valid for such nonconservative system. At the same time in the case of nonselective interaction the Robinson’s damping criterion isvalid: the sum of damping decrements (inverse damping
times) is a constant. In a particular choice of lattice andlaser or material target, damping rates can be shifted between different degrees of freedom. However the decrements are limited by the value τ y,s ≥ 2(1 + D)ε/(2 +D)P > 0 [4, 5].

2

ENHANCED LASER COOLING

The first method of enhanced one-dimensional laser cooling of ion beams in the longitudinal plane is based on theresonance Rayleigh scattering of ”monochromatic” laser
photons by not fully stripped ion beams or by complicatednuclei [6] - [10]. In this method the laser beam overlaps theion beam and has a chirp of frequency. Ions interact withthe laser beam at resonance energy, decrease their energy
in the process of the laser frequency scanning until all ofthem reach the minimum energy of ions in the beam. At
this frequency the laser beam is switched off. The higher
energy of ions the earlier they begin interaction with thelaser beam, the longer the time of interaction. Ions of minimum energy do not interact with the laser beam at all. In
such a way, in this method, the selective interaction is realized. The damping time of the ion beam in the longitudinalplane is determined by the dispersion of its energy spread
σε
2σε
.
(2)
P
According to (1), the damping times of betatronand phase oscillations in the three-dimensional radiative
method of cooling are equal to the time interval necessary
for the ion energy loss, which is about the two-fold andfour-fold initial energy of the ion, accordingly. The damping time (2) is ε/σ ε ∼ 103 times less than (1). The selectivity of interaction leads to the violation of the Robinson’s
damping criterion and open the possibility of enhanced
cooling.
Below we will consider the non-resonance methods ofenhanced cooling of ions and other particles both in thelongitudinal and transverse planes. The universal kind of
selective interaction of particles with moving in the radial
direction laser beams or media targets will be used. The
movement of the laser beam or media target in the radial direction leads to selective interaction with the particle beam.
In this case the target at some moment interacts with onepart of the being cooled beam and does not interact withthe rest [11], [12]. At the other moment it interacts withanother parts of the being cooled beam. By analogy withthe first method of enhanced cooling the interaction regionchange its position in the being cooled beam for the cooling process. Robinson’s damping criterion does not workin this case.
τ =

2.1

New enhanced cooling methods

For the sake of simplicity we will neglect the emission of
the synchrotron radiation in the bending magnets of thestorage rings, supposing that the RF system of a storage
ring is switched off and the broadband laser beams are homogeneous and have sharp edges in the radial directions.
We suppose that the jump of their instantaneous orbits
caused by the energy loss is less than the amplitude of betatron oscillations. In a smooth approximation, the movement of an ion relative to its instantaneous orbit is describedby the equation x β = A cos(Ωt+ϕ), where xβ = x−xη is
the ion deviation from the instantaneous orbit x η ; x, its radial coordinate; A and Ω, the amplitude and the frequency
of betatron oscillations. If the coordinate x β 0 and transverse radial velocity of the ion ẋ β 0 = −AΩ sin(Ωt0 +ϕ)correspond to the moment t 0 of change of the ion energy
in a laser beam then the amplitude of betatron oscillations
of the ion before an interaction is A 0 = x2β 0 + ẋ2β 0 /Ω2 .
After the interaction, the position of the ion instantaneous
orbit will be changed by a value δx η , the deviation of theion relative to the new orbit will be x β 0 − δxη , and the
direction of its velocitywill not be changed. The new amplitude will be A1 = (xβ 0 − δxη )2 + ẋ2β 0 /Ω2 and thechange of the square of the amplitude
δ(A)2 = A21 − A20 = −2xβ 0 δxη + (δxη )2 .

(3)

When |δxη |  |xβ 0 | < A0 then in the first approximation the value δA = −(xβ 0 /A)δxη . From this it follows
that to produce the enhanced cooling of an ion beam in
the transverse plane we must create such conditions whenions interact with a laser beam under deviations from theinstantaneous orbit x β0 of one sign (xβ 0 < 0, when thedispersion function is positive ∂x η /∂ε > 0 or in the opposite case xβ 0 > 0, ∂xη /∂ε < 0). In this case the valueδA has one sign and the rate of change of amplitudes ofbetatron oscillations of ions ∂A/∂t is maximum. A selective interaction of ions with the laser beam is necessary torealize this case.
Two schemes of a selective interaction of ion and laser
beams for cooling of ion beams in the transverse and longitudinal planes can be suggested. For the transverse cooling,
the laser beam T 1 is used. At the initial moment it overlaps
a small external part of the ion beam in the radial directionin the straight section of the storage ring with non zero dispersion function. First ions with largest initial amplitudes
of betatron oscillations interact with the laser beam. Immediately after the interaction and loss of the energy the
position and direction of momentum of an ion remain thesame, but the instantaneous orbit is displaced inward in the
direction of the laser beam. The radial coordinate of theinstantaneous orbit and the amplitude of betatron oscillations are decreased to the same value owing to the dispersion coupling. After every interaction the position of theinstantaneous orbit approaches the laser beam more andmore, and the amplitude of betatron oscillations is coming
smaller. It will reach some small value when the instantaneous orbit reaches the edge of the laser beam. Whenthe depth of dipping of the instantaneous orbit of the ion
in the laser beam becomes greater than the amplitude of its
betatron oscillations, the orbit will continue its movementin the laser beam with constant velocity. The amplitude ofbetatron oscillations will not be changed.
The degree of overlapping is changed by moving uniformly the laser beam position from inside in the direction of the being cooled ion beam with some velocity v T1 1 .
When the laser beam reaches the instantaneous orbit corresponding to ions of maximum energies then the laser beam
must be switched off and returned to a previous position.
1 Instantaneous orbits can be moved in the direction of the laser beam,instead of moving of a laser beam. A kick, decreasing of the value of themagnetic field in bending magnets of the storage ring, a phase displacement or eddy electric fields can be used for this purpose.

All ions of the beam will have small amplitudes of betatron oscillations and increased energy spread. Ions withhigh amplitudes of betatron oscillations will start to interact
with a laser beam first, their duration of interaction and absolute decrease of amplitudes of betatron oscillations will
be higher.
To realize the second enhanced cooling of an ion beam in
the longitudinal plane we can use a broadband laser beam
T2 located in the straight section of a storage ring with non
zero dispersion function. The radial laser beam position ismoving uniformly from outside in the direction of the being
cooled ion beam with a velocity v T2 higher than maximum
velocity of the ion instantaneous orbit deepened in the laser
beam. At the initial moment, the laser beam overlaps onlya small part of the ion beam. The degree of overlapping
is changed in such a way that ions of maximum energy,
first and then ions of lesser energy, come into interaction.
When the laser beam reaches the orbit of ions of minimum
energy then it must be switched off and returned to the previous position. In this case, the rate of the energy loss ofions in the beam will not be increased, but the differencein duration of interaction and hence in the energy losses ofions having maximum and minimum energies will be increased essentially. As a result all ions will be gathered at
the minimum energy in a short time.

2.2

Interaction of ion beams with transversely
moving laser beams

In the methods of enhanced laser cooling of ion beams theinternal and external laser beam positions are displaced in
the transverse directions. Below the evolution of amplitudes of betatron oscillations and positions of instantaneous
orbits in the process of the energy loss of ions in laser
beams will be analyzed.
The velocity of an ion instantaneous orbit ẋ η depends onthe distance xT1,2 − xη between the edge of the laser beamand the instantaneous orbit, and on the amplitude of betatron oscillations. When the orbit enters the laser beam at
the depth higher than the amplitude of betatron oscillations
then ions interact with the laser beam every turn and theirs
velocity reaches the maximum value ẋ η in which is givenby the intensity and the length of the interaction region of
the ion and laser beams. In the general case, the velocityẋη can be presented in the form ẋ η = W · ẋη in , where
W is the probability of an ion crossing the laser beam.
W is the ratio to a period of a part of the period of betatron oscillations of the ion determined by the condition|xT12 − xη | ≤ |x0 | ≤ A when the deviation of the ionfrom the instantaneous orbit at azimuths of the laser beam
is directed to the laser beam and is greater than the distance
between the orbit and the laser beam. When the length ofthe laser beam is much less than the length of the period ofion betatron oscillations, the probability can be presented
in the form W = ϕ1,2 /π, where ϕ1 = π − arccos ξ1 ,ϕ2 = arccos ξ2 , ξ1,2 = (xT1,2 − xη )/A, indices 1,2 correspond to laser beams.

The behavior of the amplitudes of betatron oscillationsof ions, according to (3), is determined by the equation∂A/∂xη = − < xβ 0 > /A, where < xβ 0 > is the ion deviation from the instantaneous orbit averaged through therange of phases 2ϕ 1,2 of betatron oscillations where ionscross the laser beam. The value < x β 0 >= ±Asinc ϕ1,2 ,where sincϕ1,2 = sinϕ1,2 /ϕ1,2 , signs + and − are related to the first and second laser beams. Thus the coolingprocesses are determined by the system of equations∂A= ±sincϕ1,2 ,∂xη

ẋη in∂xη
=ϕ1,2 .
∂tπẋη in∂Aẋη in 
2 .
=sin ϕ1,2 =1 − ξ1,
(5)
2
∂t
π
π
Let the initial instantaneous ion orbits be distributed ina region ±σx,ε,0 relative to the location of the middle instantaneous orbit xη , and the initial amplitudes of ion radial
betatron oscillations A0 be distributed in a region σ x,b,0 relative to their instantaneous orbits, where σ x,ε,0 and σx,b,0
are dispersions. The dispersion σ x,ε,0 is determined by the
initial energy spread σ ε,0 .
Suppose that the initial spread of amplitudes of betatronoscillations of ions σx,b,0 is identical for all instantaneous
orbits of the beam. The velocities of the instantaneous orbits in a laser beam ẋη in < 0, the transverse velocities ofthe first laser beams vT1 > 0, and vT2 < 0. Below we will
use the relative radial velocities of the laser beam displacement k1,2 = vT1,2 /ẋη in , where vT1,2 = dxT1,2 /dt. In ourcase ẋη in < 0, k1 < 0, k2 > 0, |k1,2 | < 1.
From the definition of ξ 1,2 we have a relation xη =xT1,2 − ξ1,2 A(ξ1,2 ). The time derivative is ∂xη /∂t =vT1,2 − [A + ξ1,2 (∂A/∂ξ1,2 )]∂ξ1,2 /∂t. Equating this value
to the second term in (4) we will receive the time derivative(6)

Using this equation we can transform the first value in(4) to the form ±sincϕ 1,2 (ξ1,2 ) = (∂A/∂ξ1,2 )(∂ξ1,2 /∂t)/(∂xη /∂t) = (πk1,2 − ϕ1,2 )(∂A/∂ξ1,2 )/[A + ξ1,2 (∂A/
∂ξ1,2 )]· ϕ1,2 which can be transformed to ∂ ln A/∂ξ 1,2 =± sin ϕ1,2 /πk1,2 − (ϕ1,2 ± ξ1,2 sin ϕ1,2 ). The solution ofthis equation is

A = A0 expxη (t − t0 ) = xT1,2 0 + vT1,2 (t − t0 )−A[(ξ1,2 (t − t0 )] · ξ(t − t0 ).

(4)

From equations (4) and the expression ∂A/∂x η =[∂A/∂t]/[∂xη /∂t] it follows:ẋη inπk1,2 − ϕ1,2
∂ξ1,2
=
.
∂t
π A(ξ1,2 ) + ξ1,2 (∂A/∂ξ1,2 ) ξ1,2A(ξ1,2 )/{A0 [πk1,2 −where ψ(k1,2 , ξ1,2 ) = − ξ1,2,0(ϕ1,2 ± ξ1,2 sin ∆ϕ1,2 )]}dξ1,2 .
The equations (8) determine the time dependence of thefunctions ξ1,2 (t − t0 ). The dependence of the amplitudes
A[ξ1,2 (t−t0 )] is determined by the equation (7) through thefunctions ξ1,2 (t−t0 ) in a parametric form. The dependenceof the position of the instantaneous orbit follows from thedefinition of ξ1,2

(9)

The function ψ(k 2 , ξ2 ) for the case k2 > 0 according to(8) can be presented in the form
ψ(k2 , ξ2 ) =
√

 1
ξ2

dx exp 1x√
1 − t2 /(πk2 − arccos t + t 1 − t2 )√d t.
πk2 − arccos x + x 1 − x2

(10)

The instantaneous orbits of ions having initial amplitudes of betatron oscillations A 0 will be deepened into thelaser beam to the depth greater than their final amplitudes
of betatron oscillations A f at a moment tf . According
to (8), tf = t0 + πA0 ψ(k2 , ξ2,f )/|xη in |, where ξ2,f =ξ(tf ) = 1. During the interval t f −t0 the laser beam T 2 willpass a way lf = |vT2 |(tf − t0 ) = πk2 ψ(k2 , ξ2,f )A0 . The
dependence ψ(k 2 , ξ2,f ) determined by (10) is presented inTable 1.
Table 1
k2
ψ

1.0
∞

1.02
13.8

1.03
9.90

1.05
6.52

1.1
3.7

1.2
2.1

1.3
1.5

1.4
1.2

Numerical calculations of the dependence ψ(k 2 , ξ2 ) on
ξ2 for the cases k2 = 1.0, k2 = 1.1 and k2 = 1.5 arepresented in Tables 2, 3, and 4, respectively. It can be presented in the next approximate form
ψ(k2 , ξ2 )C3 (k2 )ψ(

1 − ξ2),
k2 + ξ2(11)

where C3 (k2 ) 0.492−0.680(k 2 −1)+0.484(k2 −1)2 +..., ψ[(1 − ξ2 )/(k2 + ξ2 )]|k2 =1 (1 − ξ2 )/(1 + ξ2 ).

 ξ1,2± sin ϕ1,2 dξ1,2
, (7)
πk
−
(ϕ1,2 ± ξ1,2 sin ϕ1,2 )
1,2
ξ1,2,0where the index 0 correspond to the initial time. Substituting the values A and ∂A/∂ξ 1,2 determined by (7) in (6) wefind the relation between time of observation and parameterξ1,2t − t0 =πA0ψ(k1,2 , ξ1,2 ),|ẋη in |

(8)

2.3

The enhanced transverse laser coolingIn the method of the enhanced transverse laser cooling ofion beams a laser beam T 1 is located in the region (x T1 ,xT1 − a), where a is the laser beam width. The degreeof the transverse cooling of the ion beam is determined by(7). The final amplitude in this case can be presented in theform(k2 = 1.0)

Table 2
ξ2
ψ

1
0

0.5
.18

0.2
.34

1
0

.5
.16

.2
.3

1
0

-0.5
1.4

-0.8
4.39

-0.9
10.3

0
.42

-.2
.6

-1
3.7

-.5
1.0

-.8
2.1

-.9
2.8

-.6
.6

-.8
.77

-1
1

(k2 = 1.5)

Table 4
ξ2
ψ

-0.2
.72

(k2 = 1.1)

Table 3
ξ2
ψ

0
.49

.5
.12

.2
.2

0
.27

-.2
.36

their instantaneous orbits. Meanwhile, the spread of amplitudes of betatron oscillations is small and the step between
positions of instantaneous orbits is high.
Similar method of interaction of external and internal
targets with proton beams was described in 1956 by O’Neil[13, 14]. However, O’Neil considered the question of
damping of betatron oscillations of proton beams in thetransverse direction by means of motionless solid wedgeshaped material targets. The targets in that case could
not provide any enhanced cooling. They could be used
for injection and capture of only one portion of protons.
For the multi-cycle injection and storage of protons O’Neilsuggested an ordinary three-dimensional ionization cooling based on a thin hydrogen jet target located in a working
region of the storage ring.

2.4
1 − ξ12 dξ1 ξ1,f.
1 − ξ12(12)
The numerical calculations of the dependence of the ratio Af /A0 on the relative radial velocity k 1 of the laser
beam displacement are presented Table 5 for the case
ξ1,0 = −1, ξ1,f = 1. This dependence can be presented
by the approximate expression

|k1 |
.
(13)
Af A0
|k1 | + 1
Af = A0 exp

ξ1,0πk1 − π + arccos ξ1 − ξ1

Table 5|k1 |Af /A0

0
0

0.2
0.41

0.4
0.54

0.6
0.61

0.8
0.67

1.0
0.71

σx,0
,
vT1

σx,f ||k1 |1σx,0+ σx,ε,0 ,|k1 |In the method of the enhanced longitudinal laser coolingof ion beams a laser beam T 2 is located in the region (x T2 ,xT2 + a). Its radial position is displaced uniformly withthe velocity vT2 < 0, |vT2 | > |ẋη in | from outside of theworking region of the storage ring in the direction of a being cooled ion beam. The instantaneous orbits of ions will
go in the same direction with a velocity |ẋ η | ≤ |ẋη in | beginning from the moment of their first interaction with thelaser beam. When the laser beam reaches the instantaneous
orbit of ions having minimum initial energies it must be
removed to the initial position.
The law of change of the amplitudes of ion betatron oscillations is determined by (7), which can be presented in
the form− 1 − ξ22 dξ2.
A = A0 exp
1 − ξ22
ξ2,0 πk2 − arccos ξ 2 + ξ2
 ξ2

The time of the laser beam cooling and the final total
radial dimension of the beam are equal toτx,1

The enhanced longitudinal laser cooling(14)

where σx,0 = σx,b,0 +σx,ε,0 is the total initial radial dimension of the ion beam. For the time τ x,1 the instantaneous
orbits of ions of a beam having minimum energy and maximum amplitudes of betatron oscillations at |k 1 |  1 pass
the distance ∼ |ẋη in |τx,1 .
According to (13) and (14) the enhanced transverse laser
cooling can lead to an appreciable degree of cooling of ion
beams in the transverse plane and a much greater degree ofheating in the longitudinal one. From this it follows thatthe four- and six-dimensional emittances of the beam will
be increased. Straight sections with low-beta and high dispersion functions have to use. In this case the radial beamdimension (14) will be lesser and lesser events of photon
emission are required to cool the beam in the transverse
direction, as the change of amplitudes of betatron oscillations of ions is of the order of the change of positions of(15)

The dependence of the ratio of a final amplitude of ion
betatron oscillations Af = A(ξ2 = 1) to the initial one
on the relative velocity k 2 of the second laser beam is presented in Table 6. This ratio can be presented by the next
approximate expression

k2
.
(16)
Af A0
k2 − 1
Table 6k2
Af /A01.0001
100.0

1.001
31.64

1.01
10

1.1
3.3

1.5
1.7

2.0
1.4

The evolution of instantaneous orbits of ions interactingwith the laser beam depends on the initial amplitudes of betatron oscillations of these ions. First of all the laser beam
T2 interacts with ions having the largest initial amplitudes
of betatron oscillations A 0 = σx,b,0 and the highest energies. The instantaneous orbit of these ions, according to (7)- (9), is changed by the law x η1 = xT2 ,0 + vT2 (t − t01 ) −ξ2 σx,b (ξ2 ) up to the time t = t f , where t01 is the initialtime of interaction of ions with the laser beam. At the sametime instantaneous orbits x η2 of ions having the same maximum energy but zero amplitudes of betatron oscillations
are at rest up to the moment t 02 = t01 + σx,b,0 /|vT2 |. The
orbit xη1 is displaced relative to the orbit x η2 by the distance ∆xη1−2 = (xη1 − xη2 ). At the moment t02 , whenxη2 = xT2 , this distance reaches the minimum∆xη1−2 m (t02 ) = −ξ2 (t02 ) · σx,b (t02 ) < 0,

(k2 − 1)
[σx,b,0 + vT2 (t − t01 )] − ξ2 σx,b (ξ2 ) =k2−[

k2 − 1 lT2(− 1) + ξ2 D2 )]σx,b,0 ,k2
σx,b,0(18)

where D2 = D2 (k2 , ξ2 ) = σx,b /σx,b,0 , lT2 = xT2 −xT2 ,0 = πk2 ψ(k2 , ξ2 )σx,b,0 ≤ lf is the displacement of
the laser beam. The typical dependence D 2 defined by (15)is presented in Fig.1.
When t > tf then the value ξ2 = ξ2,f = −1, lT2 = lf ,D2 = k2 /(k2 − 1) and (18) have the maximumk2
k2 − 1∆xη1−2 |t>tf = [−+k2
k2 − 1π(k2 − 1)ψ(k2 , ξ2,f )]σx,b,0 .

(19)

The instantaneous orbit x η2 will be at a distance xη2−3 =[(k2 −1)/k2 ]σx,η,0 from the motionless instantaneous orbitxη3 of ions having minimum energy and zero amplitudes ofbetatron oscillations when the laser beam is stopped at theposition xη3 .
If we take into account that the instantaneous orbits ofions having maximum amplitudes of betatron oscillations
and the minimum energy are below the instantaneous orbitsof ions having zero amplitudes of betatron oscillations andminimum energy, by the value 0.28σ x,b,0 , at the momentof the laser beam stopping then the total radial dispersionof the instantaneous orbits of the beam can be presented in
the form

σx,ε,f ≤k2 − 1
σx,0 + [k2

+0.28]σx,b,0 ,k2− π(k2 − 1)ψ(k2 , ξ2,f )k2 − 1AT2 > lf , σx,0 .

10k2 = 1.01(17)

where the parameter ξ 2 (t02 ), according to (8) and thecondition |vT2 |(t02 − t01 ) = σx,b,0 , will be determinedby the equation ψ[k 2 , ξ2 (t02 )] = 1/πk2 . The valueψ[k2 , ξ2 (t02 )]|k2 11/π, ξ2 (k2 , t02 )|k2 1
0.22 (seeTables 2-4), σx,b (t02 ) = 1.26σx,b,0 and the distance
0.28σx,b,0 . This distance is decreased|∆xη1−2 (t02 )|with increasing k2 .
The instantaneous orbit of particles x η2 inside the interval t02 < t ≤ tf is changed by the law x η2 = xT2 ,0−σx,b,0 + ẋη, in (t − t01 + σx,b,0 /vT2 ) and the distance∆xη1−2 =The damping time of the ion beam in the longitudinalplane τε = 2σε,0 (1 + σx, b,0 /σx, ε,0 )/P .

(20)

D2

8

6

k2 = 1.14

k2 = 1.5
-1

2

-0.5

0

0.5
x

ξ2

1

Figure 1: The dependence on ξ2 of the ratio of a current amplitude of betatron oscillations to an initial amplitude D2 =A/A0 = σx,b /σx,b,0 .
According to (20) the efficiency of the enhanced longitudinal laser cooling is the higher the less the ratio of the
spread of the initial amplitudes of betatron oscillations tothe spread of the instantaneous orbits of the being cooled
ion beam.
According to (16) and (20) the enhanced longitudinallaser cooling can lead to a high degree of cooling of ion
beams in the longitudinal plane and a much lesser degreeof heating in the transverse one. From this it follows thatthe four- and six-dimensional emittances of the beam will
be decreased.

3 DISCUSSION
In the method of the enhanced transverse laser coolingof ion beams, according to (13) and (14), the degreeof compression of betatron oscillations C 1 = A0 /A =(1 + |k1 |)/|k1 | and the degree of decompression of the
spread of the instantaneous orbits of the beam D 1
C12 .
In the second method of the enhanced longitudinal laser
cooling, according to (16) and (20), there is a significant
decrease in the spread of instantaneous orbits of ions defined by the coefficient C 2,l = σx,ε,0 /σx,ε,f , and a lesser
value of√ increase in the amplitudes of betatron oscillations
D2
C2 . From this it follows that cooling of ion beams
both in the transverse and longitudinal planes, in turn or simultaneously, does not lead to their total cooling in theseplanes. We can cool ion beams either in the transverse or
longitudinal planes or look for combinations of these methods of cooling with other methods.
In the method of enhanced longitudinal laser cooling,
contrary to the transverse one, the degree of longitudinalcooling is greater than the degree of heating in the trans-

verse plane. That is why we can use the emittance exchange between longitudinal and transverse planes, e.g.,using a synchro-betatron resonance [15] - [18] or dispersion coupling by additional motionless wedge-shaped targets [13, 14, 19] together with the moving target T 2 and insuch a way to realize the enhanced two-dimensional cooling of the ion beam based on the longitudinal laser coolingonly2 .
When the synchrotron radiation damping of ion beams in
guiding magnetic fields of lattices of storage rings is highand the radio frequency accelerating system is switched onthen we can do an additional enhanced laser cooling of such
beams in the radio frequency buckets. Such cooling in thelongitudinal plane can be produced by using of a being displaced laser beam T 2 at the condition when losses of energy
by ions in the laser beam are higher than synchrotron radiation losses. Cooling of the ion beams in the RF buckets is
another problem to be considered elsewhere.
The enhanced transverse method of laser cooling together with the first enhanced longitudinal one can be used
for cooling of ion beams. A heating of the ion beam in thelongitudinal plane in the process of the enhanced transverse
cooling will be compensated completely by its following
cooling in the longitudinal plane.
If the laser beam has a broad spectrum with sharp frequency edges then the transverse method of cooling willlead to the longitudinal as well 3 . Ion beam will be cooledin the transverse plane and all ions will be gathered at someenergy lesser than minimal after several cycles of transverse cooling.

4 CONCLUSION
The enhanced methods of cooling can be applied to ion,
electron and muon beams. Such beams can be used in theelementary particle physics and can lead to new generations of light sources [3], [20] - [22].

5

REFERENCES[1] E.G.Bessonov, Proc. Internat. Linear Accel. Conf. LINAC94,
Tsukuba, KEK, 1994, Vol.2, pp.786; Journal of Russian
Laser Research, 15, No 5, (1994), p.403.
[2] E.G.Bessonov and Kwang-Je Kim, Preprint LBL-37458 UC414, 1995; Phys. Rev. Lett., 1996, vol.76, p.431.
[3] E.G.Bessonov, K.-J.Kim, Proc. 1995 Part. Accel. Conf.
and Int. Conf. on High-Energy Accelerators, p.2895; Proc.
5th European Particle Accelerator Conference, Sitges,
Barcelona, 10-14 June 1996, v.2, p. 1196.
[4] H.Wiedemann, Particle Accelerator Physics I(Springer-Verlag, New York, 1993).

and II[5] E.G.Bessonov, physics/0202040.
[6] P.J.Channel, J. Apply Physics, v. 52(6), p.3791 (1981).
2 Similar way a cooling of muon beams can be done by material targets.
3 We can use a ”monochromatic” laser beam with a fast periodical
modulation of frequency as well.

[7] P.J.Channel, L.D.Selvo, R.Bonifacio, W.Barletta, Optics
Communications, v.116, (1995), p.374.
[8] S.Shröder, R.Clein, N.Boos, et al., Phys. Rev. Lett., v. 64, No
24, p.2901 (1990).
[9] J.S.Hangst, M.Kristensen, J.S.Nielsen O.Poulsen,
Schifter, P.Shi, Phys. Rev. Lett., v. 67. 1238 (1991).

J.P.

[10] J.S.Hangst, K.Berg-Sorensen, P.S.Jessen, et al., Proc. IEEE
Part. Accel. Conf., San Francisco, May 6-9, NY, 1991, v.3,
p.1764.
[11] E.G.Bessonov, K.-J Kim, F.Willeke, physics/9812043.
[12] E.G.Bessonov, physics/0001067.
[13] O’Neil G., Phys. Rev., 102, 1418 (1956);[14] A.Shoch, Nucl. Instr. Meth, v.11, p.40 (1961).
[15] K.W.Robinson, Phys. Rev., 1958, v.111, No 2, p.373.
[16] A.Hoffman, R.Little, J.Peterson, Proc. VI Int. Conf. HighEnergy Accel. Cambridge (Mass.), 1967, p.123.
[17] H.Okamoto, A.M.Sessler, and D.Möhl, Phys. Rev. Lett. 72,
3977 (1994).
[18] T.Kihara, H.Okamoto, Y.Iwashita, et al., Phys. Rev. E, v.59,
No 3, p. 3594, (1999).
[19] D.V.Neufer, Nucl. Instr. Methods, 1994, v.A350, p.24.
[20] E.G.Bessonov, NIM. v.A358, (1995), pp. 204-207.
[21] E.G.Bessonov, Proc. ESRF-ICFA WS ”4th generation lightsources”, 1996, ESRF, Grenoble, France, p.WG2-108.
[22] Zh. Huang, R.D.Ruth, PRL, v.80, No 5, 1998, p. 976.