SLAC-PUB-14808 EMITTANCE ADAPTER FOR A DIFFRACTION LIMITED SYNCHROTRON RADIATION SOURCE Alexander Wu Chao* and Pantaleo Raimondi# * SLAC National Accelerator Lab., 2575 Sand Hill Road, Menlo Park, CA 94025, USA; Work supported by DoE contract D-AC02- 76SF00515. # LNF-INFN, Via E. Fermi 40, 00044 Frascati (Rome), Italy. Abstract sufficiently large to accommodate a radiator (e.g. an We investigate the possibility of reaching very small undulator) inside. After the solenoid, a round-to-flat horizontal and vertical emittances inside an undulator in a adapter (mirror image of the flat-to-round adapter) storage ring, by means of a local exchange of the apparent restores the original coupling. horizontal and vertical emittances, performed with a The arc ring lattice should provide a beam as flat as combination of skew quadrupoles and one solenoid in a possible so that for the (εy<<εx), ε xε y < λ / 4π dedicated insertion line in the storage ring. The insertion radiation wavelength of interest. leaves the ring parameters and its optical properties unaffected. This scheme could greatly relax the emittance Quantities εx and εy are the eigen-emittances of the requirements for a diffraction limited synchrotron light storage ring in the canonical phase space (x, px, y, py). The source. The lattice derivation and design is described. adapters and the solenoid are magnets whose symplectic transport maps will not affect them [12]. It is therefore not possible to make the canonical momenta px and py very INTRODUCTION small simultaneously inside the solenoid. However, in the solenoid, the mechanical momenta x’ and y’ can be made The idea of flat-to-round and round-to-flat adapters small simultaneously without violating symplecticity has been first introduced by Derbenev [1] and extended because they are not the canonical coordinates. The by him and many others [2-10]. He first envisioned adapter makes them determined solely by the small initial applying it to a storage ring collider to form round beams ε . Fortunately, radiation cares about the apparent at the collision point to minimize the effect of beam-beam y emittances in the (x, x’, y, y’) space rather than the true resonances. This idea has also been considered for emittances in the (x, p , y, p ) space. In the solenoid, they electron cooling [2,7]. The production of a very flat beam x y are related as: from a round photocathode immersed in a solenoid, x’=p - k y/2; y’=p + k x/2; k =B /(B ) (1) followed by a round-to-flat adapter lattice, has been x s y s s s ρ 0 experimentally demonstrated [9,10]. with Bs the solenoid magnetic field and (Bρ)0 the In the electron cooler application, the beam inside the magnetic rigidity of the electron beam. The adapter and solenoid optics make px cancel with ksy/2 (and py with solenoid has very small angular divergences σx’ and σy’. This beam could be used for electron beam cooling ksx/2), so x’ (and y’) will be small even when px (and py) because of its extremely cold temperature. We can use the are not. same technique in a back-to-back configuration in a storage ring, for the purpose of generating a diffraction- OPTICS DESIGN limited X-ray synchrotron light source [5]. We consider a regular ring lattice that, at its junction In order to have coherent or quasi-coherent radiation, to the flat-to-round section, has βx = βy, and αx = αy = 0. the beam emittances εx and εy have both to be less than We also assume that the lattice has dispersion suppressors λ/4π, where λ is the X-ray wavelength. This is very at its ends so that the dispersion vanishes in the insertion. difficult to achieve for εx, requiring complicated and We describe the particle motion using the canonical demanding storage ring lattice designs, powerful damping coordinates Xcan = (x, px, y, py) and later transform the wigglers, tight tolerances and orbit controls. The small description to X = (x, x’, y, y’). The particle motion is described by [6, 12] X = Va: dispersion function and the corresponding strong can sextupoles also lead to small dynamic apertures. ⎡ β x cosϕx β x sinϕx 0 0 ⎤ With the proposed scheme [5], the storage ring lattice ⎢ 1 1 ⎥ ⎢− sinϕ cosϕ 0 0 ⎥ can have relaxed requirements on εx, as long as εy can be x x ⎢ β x β x ⎥ made very small, as routinely obtained in electron storage V = ⎢ ⎥ 0 0 cos sin rings. In the straight section reserved for any insertion ⎢ β y ϕ y β y ϕ y ⎥ ⎢ 1 1 ⎥ device, a flat-to-round adapter is first inserted; after ⎢ 0 0 − sinϕ y cosϕ y ⎥ which, the beam enters a solenoid that has an aperture ⎢ β y β y ⎥ ⎣ ⎦ ! $ ! $ ⎡ β ⎤ 2! sin" 2! " sin(# +$ ) 1 β x 1 y # x x & # x x x x & ⎢ sin Δ − cos Δ sin Δ − cos Δ ⎥ ⎢ x x y y ⎥ # & # & β xks ks β yks ks 2!x cos"x 2"x !x cos(# x +$x ) ⎢ ⎥ # & # & ⎢ ⎥ a = ; Xcan = Va = β k β k # 2! sin" & # 2! " sin(# +$ ) & ⎢ 1 ks x s 1 ks y s ⎥ # y y & # y y y y & cos Δ x sin Δ x cos Δ y sin Δ y ⎢ 2 β 2 2 β 2 ⎥ # & # & ⎢ x y ⎥ 2!y cos"y 2"y !y cos(# y +$y ) M = "# %& "# %& adp ⎢ β ⎥ ⎢ 1 β x 1 y ⎥ cos Δ sin Δ − cos Δ − sin Δ ⎢ k x k x k y k y ⎥ At the entrance to the solenoid, the particle motion is ⎢ β x s s β y s s ⎥ ⎢ ⎥ described by Xcan = Ub, where U is the symplectic matrix k β k ⎢ 1 ks β x s 1 ks y s ⎥ consisting of the circular mode basis vectors [6]: ⎢− sin Δ x cos Δ x sin Δ y − cos Δ y ⎥ ⎢ 2 β x 2 2 β y 2 ⎥ " % ⎣ ⎦ $ 1 1 1 1 ' sin!+ ! cos!+ sin!! ! cos!! with Δx=ϕ+-ϕx and Δy=ϕ--ϕy. $ ks ks ks ks ' $ ' After inserting this section, a particle with initial $ k k k k ' s cos! s sin! s cos! s sin! condition Xcan = Va at the end of the regular cell will be $ 2 + 2 + 2 ! 2 ! ' U $ ' transported to an arbitrary distance z inside the solenoid: = $ ' !1 1 1 1 1 X = M M Va = M (UV )Va = M Ua $ cos!+ sin!+ ! cos!! ! sin!! ' can sol adp sol sol k k k k $ s s s s ' " 2 % $ k k k k ' $! ! cos(! +" + k z)+ ! cos(! +" ) ' $ s s s s ' ( x + x s y ! y ) ! sin!+ cos!+ sin!! ! cos!! $ ks ' #$ 2 2 2 2 &' $ ' ks $ !x sin(!+ +"x + ksz)+ !y sin(!! +"y ) ' ⎡ 2 ⎤ 2 ( ) ⎢− ε cos Δ + ε cos Δ ⎥ Xcan = $ ' ⎡ ⎤ k ( + + − − ) 2 2ε + sinθ+ ⎢ s ⎥ $ ! sin(! +" + k z)! ! sin(! +" ) ' ⎢ ⎥ $ ( x + x s y ! y ) ' ⎢ k ⎥ ks ⎢ 2ε + cosθ+ ⎥ s sin sin $ ' b = ; ⎢ ( ε + Δ + + ε − Δ − ) ⎥ k ⎢ ⎥ ⎢ 2 ⎥ $ s ' 2ε sinθ X can = Ub = !x cos(!+ +"x + ksz)! !y cos(!! +"y ) ⎢ − − ⎥ ⎢ 2 ⎥ #$ 2 ( ) &' ε sin Δ − ε sin Δ ⎣⎢ 2ε − cosθ− ⎦⎥ ⎢ ( + + − − ) ⎥ ⎢ ks ⎥ We have simply made the emittance exchanges εx à ε+ ⎢ ⎥ ks and à by the flat-to-round insertion. ⎢ ε cos Δ − ε cos Δ ⎥ εy ε- 2 ( + + − − ) ⎣ ⎦ The map Madp is chosen to satisfy the conditions: where Δ+=ϕ++θ+ and Δ-=ϕ-+θ-. In the language of βx= βy, ! + =! x-π/4, ! - =! y+π/4. It can be written as: circular modes, this is equivalent to choose envelope " 1 1 % " 2! % " 1 1 % $ 0 ! 0 ' 0 ! x 0 0 $ 0 0 ' function β = 2/ks and α = 0 (ks > 0). 2 2 $ ' 2 2 $ ' $ ks ' $ ' 1 1 k 1 1 At a distance z into the solenoid, the motion is $ 0 0 ! ' $ s 0 0 0 ' $ 0 0 ' $ 2 2 ' $ 2 2 ' described by Xcan = MsolUb: $ 2!x ' $ 1 1 ' $ 2 ' $ 1 1 ' $ 0 0 ' $ 0 0 ! 0 ' $! 0 0 ' ⎡ 2 ϕs 1 1 2 2 ϕs ⎤ 2 2 k ! 2 2 ⎢ cos sin ϕs sin ϕs sin ⎥ $ 1 1 ' $ s x ' $ 1 1 ' 2 k 2 k 2 ks!x ⎢ s s ⎥ $ 0 0 ' $ 0 0 0 ! ' $ 0 ! 0 ' # 2 2 & # 2 & # 2 2 & ⎢ ks 2 ϕs ks 2 ϕs 1 ⎥ − sin ϕs cos − sin sin ϕs M ⎢ 4 2 2 2 2 ⎥ The middle map is that of a quadrupole channel (three sol = ⎢ ⎥ 1 2 2 ϕs 2 ϕs 1 quadrupoles) with x phase advance of 3 /2 and y phase ⎢ − sin ϕs − sin cos sin ϕs ⎥ π ⎢ 2 ks 2 2 ks ⎥ advance of π. The two adjacent matrices mean that the ⎢ks 2 ϕs 1 ks 2 ϕs ⎥ o ⎢ sin − sin ϕs − sin ϕs cos ⎥ triplet channel is rotated by 45 so that the quadrupoles ⎣ 2 2 2 4 2 ⎦ become skew. With this insertion, a particle at the k z ϕs = s entrance of the insertion is transported inside the solenoid " % 2 with X (z) = M Ua. The result is: $ ! ! cos(" +# + k z)+ ! cos(" +# ) ' can sol ( + + + s ! ! ! ) $ ks ' ( + $ ' * 2 " # # % - ! $ !x cos(" x + +$x + ksz)+ !y cos(" y + +$y )' $ ks ' ! sin(" +# + k z)+ ! sin(" +# ) * ks # 4 4 & - $ 2 ( + + + s ! ! ! ) ' * - Xcan = $ ' * ks " # # % - $ 2 ' * $ !x sin(" x ! +$x + ksz)+ !y sin(" y + +$y )' - $ !+ sin("+ +#+ + ksz)! !! sin("! +#! ) ' 2 # 4 4 & k ( ) * - $ s ' Xcan (z) = $ ' * 2 " # # % - ks * $ !x sin(" x ! +$x + ksz)! !y sin(" y + +$y )' - $ ! cos(" +# + ksz)! ! cos(" +# ) ' k # 4 4 & $ 2 ( + + + ! ! ! ) ' * s - # & * - " % * ks # # - The “+” mode (“Lamor” mode) advances with the $ !x cos(" x + +$x + ksz)! !y cos(" y + +$y )' * 2 # 4 4 & - solenoid phase ksz, but the “-” mode’s phase stays still. ) , The adapter matches the flat optics of the ring arcs to the round optics of the solenoid.