Proposal for an Enhanced Optical Cooling system test in an electron storage ring

We are proposing to test experimentally the idea of Enhanced Optical Cooling (EOC) in an electron storage ring, to confirm new fundamental processes in beam physics and to open important applications of EOC in elementary particle physics and in Light Sources (LS).


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 s γε 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][2][3]. Recently a new method of EOC was suggested [4][5][6][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][9][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 bet x ψ = , 2 ( 1/ 2) p k k π + for first kicker undulator and bet x ψ = , 2 k k k π for the next ones, where k ij = 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].

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 . The spectral distribution of the first harmonic of UR for M>>1 is given by [11] 1 1 ; 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 ( ) 2 1 , 2 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 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: (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 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 , , , 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. 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 where ,0 E σ is the initial energy spread of the electron beam, P loss stands for the power losses (9), ,0 x σ is the initial radial beam dimension determined by betatron oscillations, , 0 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). x η δ~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.  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 The initial phases in ϕ 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 where , is the wavelength of betatron oscillations, C is the circumference of the ring, x v 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 , 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 , where s 0 corresponds to the longitudinal position of the pick up. So the transverse position of the particle has the form [12] ) is the deviation of the electron energy from the dedicated energy and dispersion D defined as , Dispersion D(s,s 0 ) 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=s 0 are zero. So full expression for transverse position of particle comes to form where x η describes periodic solution for dispersion in damping ring (slippage factor) and marks pure betatron part in transverse coordinate; η η′ stand for its value at location of pickup kicker. So the time difference becomes where we neglected terms responsible for the betatron oscillations (i.e. R 51 =0, R 52 =0).
In general case , 0 c l η ≠ the initial phase of an electron in the field of amplified URW propagating through kicker undulator, according to (15) The function takes into account that electron with some energy and its URW enter kicker undulator simultaneously at the phase if RF accelerating system is switched off (see Fig.2). The energy gaps between equilibrium energy positions have magnitudes given by 6 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 | , excitation of synchrotron oscillations by non-synchronous photons can be neglected then the electron energy is drifting to the nearest energy value m E . 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 m E 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 ,

VARIANTS OF OPTICAL COOLING
Depending on the local slippage factor and coefficients R 51 , R 52 and R 56 in (14), different variants of optical cooling can be suggested. π . 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 2. The scheme of OSC can be used at 0 , = l c η [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 low ≅ ⋅x and changes the sign at 0 x = , where G stands for the gradient. The phase of the emitted URWs changes its value on π at 0 x = 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 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 deceleration again (the phases of URWs change their value on π at 0 x = 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 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]. can be used to decrease the energy gap for cooling process and to increase the rate of cooling. , 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 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 corresponding to the range of phases and by overlapping the center of this time interval with synchronous particle, where , RF f is the frequency of the RF accelerating system of the ring. The last condition is equivalent to , where is the initial length of the electron bunch. Above we suggested that electrons are moving along elliptical trajectories 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) 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 0 x η ≠ 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].

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 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 .
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 D 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 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. 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.

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 R is the equilibrium averaged radius of the storage ring [16].
For small synchrotron oscillations the equilibrium length of the electron bunch is , , The maximum deviation of the energy from its synchronous one is where is the storage ring slippage factor of the ring.  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 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 (  Table 3) per electron in the frequency and angular ranges (3)  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 , 18.5 x EOC τ = 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 0.061 ampl P = 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 (onephoton/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 C 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 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 (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 onedimensional 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 x E A ϕ 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 Variable in time optical delay-line can be used to change in situ the length of the light passway between the undulators during the cooling cycle to move the initial phases to 2 in m /2 ϕ 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 , 0.57 Note that in the case of two-dimensional EOC using one kicker undulator, according to (18) eV determined by the nonsynchronous 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 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.

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 Amplifiersuggested 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 4 , 5 10 e N 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 9.3 ph N 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.  16.06     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 we can transform (36) to (6)

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, (  (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.

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 p ω ) amplifies a lower frequency, lower intensity beam (the signal beam, at frequency s ω ); in addition a third beam (the idler beam, at frequency i ω , with i s p ω 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 i ω ω > ).. (the so-called degeneracy condition) to p ω , and correspondingly the idler varies from 2 p ω 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.

In the interaction, energy conservation
The signal and idler group velocities s v 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 The noise (amplified self emission) power of the optical amplifier is determined by the expression , ,0 0 n n P P G = where is the noise power at the amplifier front end, is the gain of the amplifier.
,0 n P 0 G 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, .