Phase Space Gymnastics

Transcription

1 Phase Space Gymnastics As accelerator technology advances, the requirements on accelerator beam quality become increasingly demanding. Phase space gymnastics becomes a new focus of accelerator physics research. In a phase space gymnastics, the beam's phase space distribution is manipulated and precision-tailored to meet the required beam qualities. On the other hand, all realization of such gymnastics have to obey accelerator physics principles and technological limitations. Recent examples of phase space gymnastics include Emittance exchanges Phase space exchanges Emittance partitioning Seeded free electron lasers Steady-state microbunched storage rings.

2 Liouville theorem is the root cause of this phase space technology. Knowledge of the concept of phase space and its fundamental properties is assumed. Adapters The idea of adapters was first suggested by Derbenev [1993]. Consider the 4D canonical phase space X can = (x, p x, y, p y ) We have two representations to describe particle motion in this phase space. 1. For uncoupled case, use the Courant-Snyder basis of planar modes (x and y modes): where X can = Va (not to be confused with X out = M X in!)

3 2. For a fully coupled beam with rotational symmetry (e.g. in solenoid), one can use the basis of circular modes (left-handed and right handed modes) [Burov] where X can = Ub

4 Once we have the planar basis V and the circular basis U --- both are symplectic --- we are now in a position to consider ``adapters''. Here we consider transverse phase space only. Adapters can also be applied to transverse-longitudinal coupled systems.

5 Flat-to-flat adapters Flat-to-flat adapter from s 1 to s 2 is well known. The job is to design a lattice that provides the map from V(s 1 ) to V(s 2 ). The needed adapter map is This is of course a well known result. The map from s 1 to s 2 is factorizable! Note the factorization X can (s 2 ) = V(s 2 ) V(s 1 ) -1 a, depending only of (optics at s 2 ), (optics at s 1 ), (particle), respectively.

6 Round-to-round adapters Round-to-round adapter from s 1 to s 2 is given by U(s 2 )U(s 1 ) -1. The result can be written as where Two ways to accomplish this desired map: 1. A quadrupole channel that provides the map T, followed by rotating the entire subsequent beamline by. 2. A uniform solenoid with strength k s and length L will produce this map with - = = ksl/2, 1 = 2 = 2/ks, 1 = 2 =0.

7 Round-to-flat adapters A round-to-flat adapter is given by the map V(s 2 )U(s 1 ) -1, which can be shown to have a general form of a round-to-round adapter, followed by a round-toflat insertion with map(vu -1 ) 0, followed by a flat-to-flat adapter. The map (VU -1 ) 0 has a form It represents a skew quadrupole channel. Design of the adapter therefore reduces to a regular lattice matching problem. A round beam with left-handed and right-handed emittances of +, - is transformed to a planar beam with x = +, y = -.

8 Flat-to-round adapter Reversing the round-to-flat adapter, a flat beam with emittances x, y is transformed to a round beam with emittances + = x, - = y. This adapter can also be achieved by three skew quadrupoles. Applications of flat-to-round & round-to-flat adapters Storage ring colliders [Derbenev] A planar flat beam in regular arc cells is transformed by a flat-to-round adapter to become a round beam at the collision region. The collision region is immersed in a solenoidal field. After the collision region, the beam is brought back to the regular arc by a round-to-flat adapter. With a round beam at the collision point, this possibly reduces the beam-beam effect due to much reduced number of nonlinear resonances. Linear colliders [Brinkmann] A round beam is produced at the cathode immersed in a solenoidal field. After exiting the solenoid, a round-to-flat adapter transforms the beam into a flat planar configuration. The use of adapter here avoids the need of a damping ring.

9 Electron cooling [Derbenev, Burov] Applying a flat-to-round adapter to a very flat beam, a round beam can be produced with very small apparent emittances in both x and y. Immersing the beam in a matched solenoid with appropriate magnetic field, the beam becomes extremely laminar. All particles move almost straight ahead along the solenoidal field with zero Larmor radius, i.e. almost zero temperature. This is an ideal beam for performing electron cooling. Diffraction limited synchrotron radiation [Brinkmann, Raimondi, Chao] By an insertion with the configuration (flat-to-round adapter + solenoid + round-to-flat adapter), a conventional 3rd generation synchrotron radiation storage ring can in principle reach diffraction limit for X-rays without the ``ultimate ring'' design.

10 Emittance Adapter for a Diffraction Limited Synchrotron Radiation Source We investigate the possibility of reaching very small horizontal and vertical emittances inside an undulator in a storage ring, performed with an insertion of adapters. The insertion leaves the ring s optics unaffected. [Brinkman, Raimondi, Chao] If successful, this scheme relax the emittance requirements for a diffraction limited light source. However, application of the radiation from this beam meets a problem of phase space matching with the subsequent photon optics [Chavanne]. We want x < /4 and y < /4 But we have only x >> /4 and y << /4 Can we use the idea of adapters to make an ordinary ring to make diffractionlimited radiation? In the straight section reserved for any insertion device for radiation, a flat-to-round adapter is first inserted; after which, the beam enters a solenoid that has a radiator (e.g. an undulator) inside. After the solenoid, a round-to-flat adapter restores the original flat beam configuration. The remaining of the storage ring lattice should provide a beam as flat as possible ( y << x), so that ( x y) 1/2 < /4.

11 Quantities x and y are the eigen-emittances of the storage ring in the canonical phase space (x, p x, y, p y ). The adapters and the solenoid are symplectic and will not affect them. It is not possible to change the eigenemittances of the canonical coordinates. However, in the solenoid, the mechanical momenta x' and y' are not canonical coordinates, And fortunately, radiation cares about the apparent emittances in the (x, x, y, y ) space, not the (x, p x, y, p y ) space.

12 Invariance of eigen-emittances Consider a beam moving in a 3D symplectic linear system. Let the beam's second moment distributions be described by at position s=0. Let the beam be transported to another position s with a symplectic map M, i.e. We have It follows that which means the three eigenvalues of the matrix S are invariant under the transport of the beam from s=0 to s. The proof above does not require the matrix M to be the map from s=0 to arbitrary position s. It only requires M to be symplectic. Had we chosen M to be the map that brings the beam distribution to its eigenmode coordinates (x I, p I, x II, p II, x III, p III ), we can make to like

13 The six eigenvalues of eigenmode S form three pairs and are easily found to be related to the eigen-emittances, It follows then that these are also the eigenvalues of the matrix S. Since the eigenvalues of S do not change when the beam is transported, the three eigen-emittances are invariants. No matter how cleverly one designs his linear beam optics, as long as the beam line is symplectic, there will be no way to alter the eigenemittances of the beam. The best one can do is to switch them around, i.e. one can make emittance exchanges, but not emittance changes. In case one must alter the eigen-emittances, he will have to implement some beamline element that is of a nonsymplectic nature.

14 Optics design (regular cells) (flat-to-round adapter) (solenoid of arbitrary length and radius) (round-to-flat adapter) (back to regular cells) [Raimondi] Regular cells Consider a regular cell lattice that has x = y, x = y =0, =0, =0 at its ends. There is no loss of generality here. The cells only job is to produce x = 300 pm, y = 0.3 pm. They are otherwise arbitrary.

15 Solenoid Solenoid optics is described by the circular mode basis vectors with one set of - and -function, but two phases (right handed and left handed). The lattice we will match to will have = 2/k s = constant, =0, + = x /4 and - = y + /4. Once matched, propagating a distance z in the solenoid, the particle motion is described by The + mode advances with the solenoid phase s = k s z, but the - mode s phase stays still. The + mode is the Lamor mode.

16 Flat-to-round adapter The flat-to-round adapter is the key for this optics matching. Proper matching the beam must have a constant size through the solenoid ( = 2/k s = constant). After inserting this matching section, the particle with initial condition é ù 2b x e x sin(j x +q x ) 2e x b x cos(j x +q x ) X can = Va = 2b y e y sin(j y +q y ) 2e y b y cos(j y +q y ) ë û at the end of the regular cells will be transported to a distance z in the solenoid with We have simply made the emittance exchanges x + and y - by the flatto-round insertion.

17 Realization of the adapter optics The adapter map can be found to be This is just a channel of three skew quads. The beam is made to rotate by the skew quads. The rotation is exactly canceled by the entrance fringe field of the solenoid so that all particles move along the direction of solenoidal field. Upon exit, the rotation induced by the solenoid s exit fringe field will be canceled by the downstream skew quads. Round-to-flat adapter Mirror image of the flat-to-round.

18 Beam in the solenoid The radiator inside the solenoid cares about X = (x,x,y,y ) and not X can = (x,px,y,py), X = M exit X can where We obtain M exit ks ks We see that only y enters x and y! It is noted however that x and y are still determined by x. The beam therefore has a large transverse dimension (~30 m) but all particles move in the forward direction with small divergence. The beam is almost perfectly laminar.

19 The beam at the end of the regular cells has é S 0 = ë b x e x e x b x b x e y e y b x ù û The Sigma matrix inside the solenoid in the X can coordinates is é e x +e y ù k s 2 (e -e ) x y k 0 s 4 (e x +e y ) 1 2 (e x -e y ) 0 S 1 = (e e x -e y ) x +e y 0 k s (e k x -e y ) 0 0 s 4 (e x +e y ) ë û Both 0 and 1 have eigen-emittances equal to x and y, as they should.

20 In the X coordinates, we then transport 1 by M ext to obtain T S 2 = M exit S 1 M exit é = ë (e x +e y ) k s 0 0 e y 0 e y k s -e y 0 0 -e y (e x +e y ) k s 0 e y 0 0 e y k s ù û s x =s y = (e x +e y ) / k s and s x' = s y' = e y k s inside solenoid The projected beam emittance x x = [( x + y) y] 1/2 ~ ( x y) 1/2. The Diffraction limit requirement will be: That is much easier to achieve than ( x y) 1/2 < x <

21 A design example [Raimondi] Adapter insertion with all quadrupoles upright and no solenoid. Solenoid of 4m with ks=2/ =0.333/m, is to be inserted in the middle.

22 Tracking of cos-like and sin-like beam trajectories through the adapter for incoming beam displacements x=10-4 *sqrt( x ), x =10-4 /sqrt( x ), y=10-5 *sqrt( x ), y =10-5 /sqrt( x )

23 Matching the laser optics In principle, the scheme could be applied to conventional 3rd generation storage rings to provide diffraction limited X-rays. In practice, however, a severe limitation is missing a way to match the electron optics to that of the laser optics [Chavanne] Consider a single electron radiating wavelength from an undulator of length L. Although a single electron is a point charge, its radiation has a diffracted phase space area with a diffraction-limited emittance The effective -function is

24 The phase space distribution of the actual radiation needs also to convolute with the distribution of the electron beam, with a similar expression for the vertical emittance. To maximize the radiation brightness, we aim to minimize x,tot and y,tot for given electron emittances x,y. This is accomplished by x = y =. However, in the round beam scenario, the electron beam has functions equal to the Rayleigh length, To match the electron beam and the photon radiation, it is required too strong a solenoid field and/or too long a solenoid length. Some room of optimization still allows improving the photon brightness compared with the nominal cases, but as it stands, the optics matching severely limits the advantage of an adapter application to radiation sources.