SLAC-PUB-14808

EMITTANCE ADAPTER FOR A LIMITED 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 ) 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) , 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 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 εy à ε- by the flat-to-round insertion. ⎢ ( ε + cos Δ + − ε − cos Δ − ) ⎥ ⎣ 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. It is designed to provide BEAM CHARACTERISTICS IN THE -1 a transport map Madp=UV , which is: SOLENOID The radiation at the radiator cares about X and not Xcan , and X inside the solenoid is X(z)= Mexit Xcan(z) with: ⎡ 1 0 0 0⎤ The two “projected-emittances” using Σ2 are equal and ⎢ 0 1 − k 2 0⎥ M = s found to be 2 which is approximately equal to exit ⎢ 0 0 1 0⎥ !x!y +!y ⎣⎢ks 2 0 0 1⎦⎥ !x!y . The Diffraction limit requirement will be: ⎡ 2 ⎛ π π ⎞⎤ ⎢− ⎜ ε x cos(ϕ x + +θ x + ks z) + ε y cos(ϕ y + +θ y )⎟⎥ ! ! < " / 4# ⎢ ks ⎝ 4 4 ⎠⎥ x y ⎢ π ⎥ 2k ε sin(ϕ + +θ ) This is easier to achieve than! < " / 4# since today most ⎢ s y y 4 y ⎥ x X (z) = ⎢ ⎥ storage rings light sources routinely reach < 10-3 . 2 ⎛ π π ⎞ εy εx ⎢ ε sin(ϕ − +θ + k z) − ε sin(ϕ + +θ ) ⎥ ⎢ ⎜ x x x s y y y ⎟ ⎥ It should be pointed out that, since the beam is very ks ⎝ 4 4 ⎠ ⎢ ⎥ π laminar, the X-ray optics to match it will have a long ⎢ − 2k ε cos(ϕ + +θ ) ⎥ ⎢ s y y 4 y ⎥ ⎣ ⎦ Rayleigh length [13]: LR ≈ σ x σ x' ≈ ε x ε y ks . The Only the vertical emittance enters x’ and y’, as εy transverse properties of the light generated by desired, while x and y are large, mainly determined by εx. such beam have to be further studied. The beam’s appearance has a large transverse dimension The electron motion and the light properties in the but extremely small divergence. The beam is almost presence of the combined solenoid and undulator will perfectly laminar. require more detailed analysis. At the lowest order, the We now calculate the Σ matrix of the second moments orbit distortion caused by the radiator should vanish of the beam’s transverse distribution. The beam’s Σ0 at the outside of the solenoid. The first order optics remains the end of the regular cells is: same as without the radiator. " ! " 0 0 0 % $ x x ' A DESIGN EXAMPLE $ 0 "x !x 0 0 ' !0 = $ ' All lattice designs, including regular cells and adapters $ 0 0 !x"y 0 ' $ ' are straightforward and should not impose technical 0 0 0 "y !x # & challenges. The main design issue is the undulator inside The total transport map from the end of regular cells to the solenoid. Key parameters are the solenoid strength ks, its length L and physical aperture R. L and R are only inside of solenoid is MsolMadp. With this map, the Σ0 constrained by the needs of the radiator and not from matrix is transformed to (in the Xcan coordinate): # & optics matching. !x +!y 1 % 0 0 " (! "! )( Choice of ks is related to the beam characteristics at k 2 x y % s ( the radiator. In addition one should note that the skew % ( k 1 % 0 s ( ) ( ) 0 ( quadrupole strengths depend on ks. In general, extreme !x +!y !x "!y % 4 2 ( values of ks should be avoided. Typically, ks should be in !1 = % ( 1 !x +!y the range [0.1 m, 1.0 m]. % 0 (!x "!y ) 0 ( % 2 ks ( Figure 1 shows an example of the adapter design for % 1 k ( the SuperB case. On both ends, a dispersion suppressor % s ( " (!x "!y ) 0 0 (!x +!y ) cancels the arc dispersion and a triplet matches the $% 2 4 '( envelope functions for the adapter. In this example we Note that M depends on z but the beam distribution sol choose = = 6.0 m, = = 0. Subsequently there inside the solenoid is independent of z. This results from βx βy αx αy are two adapter triplets, each one generating the following the optics matching by the adapter. The beam is optically transfer matrix: “matched” into the solenoid. ⎡ π π ⎤ Eigen-emittances can be calculated from the Σ cos(ϕ + ) β sin(ϕ + ) 0 0 ⎢ 4 x 4 ⎥ matrices; they are just the eigen-values of the matrix iJΣ, ⎢ 1 π π ⎥ ⎢− sin(ϕ + ) cos(ϕ + ) 0 0 ⎥ where J is the symplectic form. Both Σ0 and Σ1 have eigen- β 4 4 V = ⎢ x ⎥ emittances equal to εx and εy. ⎢ π π ⎥ ⎢ 0 0 cos(ϕ − ) β x sin(ϕ − )⎥ In the X coordinates, we then transport Σ1 by Mexit: ⎢ 4 4 ⎥ 1 π π # & ⎢ 0 0 − sin(ϕ − ) cos(ϕ − ) ⎥ (!x +!y ) ks 0 0 !y ⎢ β 4 4 ⎥ % ( ⎣ x ⎦ % 0 ! k "! 0 ( T y s y We have set ϕ =π/2 as an example, but its value is !2 = Mexit !1Mexit = % ( % 0 "!y (!x +!y ) ks 0 ( arbitrary. In general the incoming parameters β and α in % ( the adapter section are arbitrary as well, provided that % !y 0 0 !yks ( $ ' they are equal in the two planes. From Σ2 we obtain the beam distribution moments: ! = ! = (" +" ) / k and! = ! = " k x y x y s x' y' y s inside the solenoid, all independent of z. After the solenoid, the matched round beam is brought back to its flat configuration with a reversed round-to-flat adapter, so the insertion is transparent to the ring. x 10−4 ; x' 10−4 / ; y 10−5 ; y' 10−5 / = β x = β x = β x = β x . This optic is just an example and more studies are necessary to further optimize its design and flexibility (e.g. shortening of the insertion length). However it has been shown that it is not complex and can be realized with standard magnets of moderate lengths. The only element that requires careful engineering is the undulator inside a (possibly superconduting) solenoid. SUMMARY The emittance adapter greatly relaxes many of the technical challenges of the current approaches to achieve fully transverse coherent X-ray radiation. The storage ring Figure 1: Coupled β-funtions through the insertion. The could be simpler, less technically demanding and less adapter consists of the solenoid (in the middle) and two costly than the ultra small εx designs (“ultimate” rings skew triplets on its sides. [14]), as the requirement for εx could be relaxed by a

The first adapter triplet (to the left of the solenoid in factor: 1 2 !x !y . o Fig. 1) is tilted by 45 clockwise and the second one by The scheme could be applied to conventional 3rd o 45 counter-clockwise. The 4-m solenoid, with ks = 2/βx = generation storage ring facilities (already existing or new) 0.333/m, is in the middle. The beam sizes inside are: to provide coherent soft X-rays. In an “ultimate” ring, this

! x = ! y ! 3"x and ! x' = ! y' = 0.333"y scheme could be used to reach coherent hard X-rays. In

The triplet quadrupoles are 0.5 m long and have strengths the case of a modest storage ring with εx ~10 nm, such an 0.906/-1.094/0.906 m-1 respectively. The solenoid has a insertion could generate coherent light for EUV field of 2.8 Tesla for a 4 GeV beam. lithography at λ = 13.5 nm. The effectiveness of the scheme is shown in Fig. 2 where cosine-like and sine-like beam trajectories are ACKNOWLEDGEMENTS tracked through the adapter. In the general case both We would like to thank M. Biagini, M. Boscolo, Y. incoming x and x’ become pure x and y in the solenoid. Cai, S. Derbenev, Z. Huang, R. Hettel, Y. Jiao, D. Ratner, For the chosen case ϕ =π/2, x becomes a pure y D. Robin, J. Safranek, C. Steier, G. Stupakov, M. displacement and x’ a pure x displacement. Incoming y Sullivan, and D. Xiang for many illuminating discussions. and y’ are the only contributions to the horizontal and vertical angles in the solenoid. Such angles are very small REFERENCES -2 since we have assumed εy = 10 εx. This behaviour is [1] Y. Derbenev, Michigan Univ. Report No. 91-2 exactly as the formulas above described. We also stress (1991); UM HE 93-20 (1993); Workshop on Round that such condition happens just in the solenoid; indeed Beams and Related Concepts in Beam Dynamics, the angles due to εx vanish as soon as the beam enters the Fermilab (1996). solenoid. The coupling disappears after the insertion and [2] Y. Derbenev, Michigan Univ. Report UM HE 98-04 the beam stays flat in the rest of the ring. (1998). [3] A. Burov and V. Danilov, FNAL Report No. TM- 2040 (1998); FNAL Report TM-2043 (1998). [4] R. Brinkmann, Y. Derbenev, and K. Floettmann, Phys. Rev. Special Topics – Accel. & Beams, Vol. 4, 053501 (2001). [5] R. Brinkmann, Proc. Euro. Part. Accel. Conf., TUPRI044 (2002). [6] A. Burov, S. Nagaitsev, and Ysroslov Derbenev, Phys. Rev. E 66, 016503 (2002). [7] A. Burov, S. Nagaitsev, A. Shemyakin, and Ya. Derbenev, Phys. Rev. Special Topics – Accel. & Beams, Vol. 3, 094002 (2000). [8] Kwang-Je Kim, Phys. Rev. Special Topics – Accel. & Beams, Vol. 6, 104002 (2003) Figure 2: Tracking through the adapter for incoming [9] D. Edwards et al., Proc. 2001 Part. Accel. Conf., -2 beam displacements (εx = 10 nm, εy = 10 εx): Chicago, IL (IEEE, Piscataway, NJ, 2001), p. 73. [10] P. Piot, Y.-E. Sun, K.-J. Kim, Phys. Rev. Special Topics – Accel. & Beams, Vol. 9, 031001 (2006). [11] A. Dragt, F. Neri, G. Rangarajan, Phys. Rev. A 45, 2572 (1992). [12] E.D. Courant, H.S. Snyder, Ann. Phys. 3, 1 (1958). [13] A.M.Kondratenko and E.L.Saldin, Dokl. Akad. Nauk. SSSR v.249, p.843 (1979); Part. Accel. v.10, p.207 (1980); Zhurnal Tech. Fiz. v.51 p.1633 (1981). [14] See, for example, M. Bei et al., to be published in Nucl. Inst. Meth. In Phys. Res. A (2011).