Idea proposed by M. Cornacchia (Nov. 2001)

Transcription

1 rnsverse to Longitudinl Emittnce Echnge M. Corncchi, P. Emm My 17, 00 Ide proposed by M. Corncchi (Nov. 001) Anlysis tken from similr work by W. Spence nd PE (unpublished, ~1990) Motivtion to reduce trnsverse emittnce nd increse longitudinl emittnce fster SASE X-ry FEL lsing nd less CSR micro-bunching in compressors

2 X-ry SASE FEL needs emittnce λ r γε < γ, 4 π nd energy spred σ δ 13 I pk λ u K π IA βε γ 1 1 < ρ 4 N At λ r 1 Å nd γ γε < 0. µm But σ δ < 0.09% (for I pk 4 ka, λ u 3 cm, K 4, β 0) his mens γσ σ δ = γε < 500 µm RF-guns produce γε ~ 5 µm nd γε ~ µm (nd highly vrible) Cn we echnge these in some wy, reducing ε while incresing ε (mybe lso dmping the CSR microbunching)?.

3 Initil uncoupled 4 4 bem covrince mtri σ 0 [γ = (1+α )/β, γ = (1+α )/β ] σ 0 ε β 0 ε α 0 ε α ε γ ε β ε α ε α 0 ε γ 0 σ 0 = 0 σ = σ = ε β, σ = ε β δ α chirp: =, ( δ EE0) σ β ( ) E-spred: σ 1 δ= ε + α β = σδ + σδ = δ γε γ σ σ δ E ( = 0) = δ = γε α γσ σ σ mc σ u c

4 Propgte σ 0 through 4 4 trnsfer mtri, R 0 σ= Rσ R, which is four blocks A B R=, C D 11 1 b11 b1 A=, =, etc. B b b 1 1 AσΑ+ BσB AσC + BσD σ= Cσ Α Dσ B Cσ C Dσ D = AσΑ+ Bσ B = CσC + Dσ D Determinnt of sum of mtrices { } X+ Y= X+ Y+tr X Y, where X is the djoint of X

5 1 X = XX, X 0, or 1 X = J X J nd J is 0 1, J J = I. 1 0 he emittnces re {( ) } = + + tr ε A ε B ε Aσ A Bσ B {( ) } = + + tr ε C ε D ε Cσ C Dσ D Now use lternte form for σ, 1 β 0 σ= ε QQ, 0 Q, β α 1 nd lso use property of trce tr{ XYZ} = tr{ YZX} = tr{ ZXY} Emittnces re then,

6 where = + + = + + { } ε A ε B ε ε ε tr UU { } ε C ε D ε ε ε tr VV 1 1 U Q A BQ V Q C DQ Now use the symplectic condition J 0 RSR= RSR = S= 0 J A JA C JC A JB C JD RSR B JA + D JC B JB + D JD RSR + + = = AJA + BJB AJC + BJD = CJA + DJB CJC + DJD A JA+ C JC = AJA + BJB = 1, B JB+ D JD = CJC + DJD = 1,.,,

7 nd find the reltions between the submtri determinnts A+ C=± 1, A= D, B= C, nd lso between U nd V ( ) V Q C DQ = Q J C J DQ, nd using from bove herefore, CJD= AJB, V Q J ( A JB) Q = Q A BQ = U tr { } { UU = tr VV } which is the sum of squres of the normlied coupling mtri, nd is positive. { } UU tr = U + U + U + U λ he emittnces re

8 ( A) = A = ( A) + A + ε ε 1 ε ε ε λ, ε 1 ε ε ε ε λ. Now introduce rectngulr trnsverse RF cvity operting in M 110 mode B E y V0 V0 = cos ωt, ) ) V V = sinωt ) ω ). c

9 rnsverse RF ccelertes nd kicks bem ev0 ev0 δ = k, = k E E he trnsfer mtri of this thin-lens cvity is k 0 R k =, 1 0 k 1 which looks like thin-lens skew qud, but here we consider the spce (,,, δ). Now plce this cvity in chicne

10 R k R 1 R R R R 56 ρ L 1 R= R R R k 1 1 L/ 0 η 0 1 = 0 η 1 ρ/ L/ 0 η 0 1 = 0 η 1 ρ/ 0 1

11 ρl 1 + ηk L kl/ k η ηk k kρ/ R = ρl kρ/ k η 1+ ηk ρ 4 k kl/ 0 1 ηk A= D= 1 η k, B= C = η Now work out λ λ { UU } 1 1 U= Q A BQ 1 = tr = 4 terms, or for k =... η λ Emittnce re ( + )( ) + δ ββ ε ε = α α = σ σ η. k k

12 = + 4 δ > ε ε σ σ η ε = + 4 δ > ε ε σ σ η ε If ε 0 = ε 0, then ε = ε (i.e., lwys equl). For emple, tke β =.6 m, σ = 100 µm, σ E = 5 kev, β =.9 m, α = α = 0, E 0 = 150 MeV, η = 100 mm, k = 10 m 1, = 1 cm, V 0 = 15 MV γε = 5 µ m γε = γε µ m γε = 1 µ m γε = γε µ m Get nerly complete emittnce echnge.

13 Bunch length for k = 1/η, nd α = 0 σ ε ρ σ Bunch lso typiclly compresses. ρβ 1 ρl 1 = k + δ 4 β + 4 k Second-order optics ( 166 ) cn be significnt, but use smll β nd/or lrge η (smll k) for control

14 β = 10 m, η = 50 mm σ E /E 0.8% γε 3.1 µm (liner: 1 µm). γε 10 µm (liner: 5 µm). But, for β =.6 m, η = 100 mm σ E /E 0.% get γε 1.07 µm γε 5. µm

15 olernces 1 σ centroid jitter becomes 1 σ jitter, but in the other plne: 0 = σ 0 = σ, δ = σ δ 0 = σ 0 δ σ δ / 0 = σ 0 = σ, = σ δ 0 = σ δ0 = σ /30 V/V 0 < 0.5% ε/ε < 3 % φ < 0.3 S-bnd < σ, < σ α <.5 (i.e., projected energy spred up to 3-times lrger thn intrinsic spred, for these prmeters) ε/ε < 10 %