Analysis of spontaneous emission and its self-amplification in free-electron laser

Transcription

1 FLS006 DESY Analyi of pontanou miion and it lf-amplification in fr-lctron lar Jia Qika ( 贾启卡 ) 18 May 006 National Synchrotron Radiation laboratory Univrity of Scinc and Tchnology of China Hfi, Anhui, 3009, China jiaqk@utc.du.cn

2 FLS006 DESY FEL quation in th tim domain, gain dcribing, th caling and FEL imulation invoking a avrag of th - bam ovr a longitudinal ditanc not applicabl to th problm rlatd pontanou miion Spontanou miion tudid with th L.-W potntial quation gdfhhhhhhhhhhhhhhhh hhdo not dcrib th intraction of -_ radiation thory tudy of SASE i carrid out in th frquncy domain Including th pontanou miion in th am framwork i maningful and convnint to FEL analyi with th tim domain approach: w invtigat pontanou miion (incohrnt and cohrnt) and it lf-amplification in FEL

3 FLS006 DESY Spontanou Emiion Equation 1 λ raδ Σ u p i( kz ωt) ikuz j ( + ) a δ ( z zj) z c t γ j δ p 1, circular polarization [J,J] n, n1,3,5,... lin polarization.. chang th variabl (z, t) jth - z j βct+z 0 +ζ j βτ+z 0 +ζ j, Z z ct, τ ct z 0 initial poition of th rfrnc point ζ j rlativ poition rpct to z 0 within th bunch for jth -. λ raδ + 0 ik ( z + βτ+ ξ ) u p ikz u 0 j a δ( S ( ζ z Z)) τ γσ j S(1-β)τ : lippag ditanc. j

4 FLS006 DESY th coordinat variabl rlation Undulator ntranc

5 FLS006 DESY ζ j : Gnrally i dpndnt on th tim but for th pontanou miion - hav no intraction with optical fild, i indpndnt on th tim Spontanou miion φ ku r i Z i ( j z0 ) λuauδ ζ + p 1 β 1 β p τ 0 ζ j 0 ζ j γσ j a ( Z, ) H( S z + Z) H( z + Z) rλuauδp ik (, ) ζ j a p Z τ H( S ζ j z0 + Z) H( ζ j + z0 Z) γσ j < > : th nmbl avrag ovr bunch

6 FLS006 DESY incohrnt pontanou miion: (trm ij) a rλ a δ ( ) u u p SE γσ N,l th numbr of - in th ditanc lζ -ζ 1 N, l l / < ζ < l / b j b ζ 1 max[ lb /, Z z0] ζ min[ l /, S+ Z z ] b 0 cohrnt pontanou miion: (cro trm i j ) N rλuauδp ik( ζ j ζl) CSE τ ζ j 0 ζ j 0 γσ jl, 1( j l) a ( Z, ) H( S z + Z) H( + z Z) * H( S ζ z + Z) H( ζ + z Z) l 0 l 0 rλuauδp ikζ N f( ζ ) H( S ζ z0 + Z) H( ζ + z0 Z) dζ γσ f(ζ) : normalizd - dnity ditribution function

7 FLS006 DESY Writtn togthr rλ a δ ζ ζ p + a Z N f d N f d u u p ikζ (, τ) ( ζ) ζ ( ζ) ζ γ Σ ζ1 ζ 1 Both incohrnt pontanou miion and cohrnt pontanou miion rlatd with lippag ditanc in th body of th radiation pul l for hort - bunch (l b b << S) lζ -ζ 1 S for long - bunch (l b >> S)

8 FLS006 DESY long - bunch hort - bunch l b 1,S4 SE l b 4, S1 SE ~ 0. ~ CSE(*100/N ) CSE(*100/N ) Z Z Incohrnt and cohrnt pontanou miion for a rctangl profil - pul Th longitudinal quantiti ar cald to th radiation wavlngth

9 FLS006 DESY long - bunch hort - bunch ~ l 0.5 b 30, 3 CSE(*10 4 /N 0.0 SE ) Z ~ l b 4,1 SE CSE(*100/N ) Z Incohrnt and cohrnt pontanou miion for a Gauian profil - pul Th longitudinal quantiti ar cald to th radiation wavlngth

10 FLS006 DESY Som intanc rctangl - bunch ditribution: owr kl N 3 SE ( ) ρ /, l k l 3 CSE 16ρ in ( / ) CSE SE π N in c ( l) N l, l / λ 1 l, << λ Enrgy l: l(z), intgral ovr Z W k l S N 3 SE 4 b ρ / W S l 3 CSE 16ρ min[, b] W CSE λ min[ S, lb ] N WSE πlb S

11 FLS006 DESY for a long - bunch (l b >>S) coating bam in th body of th radiation pul it ha ls a 16 πnλ rγρ / Σ 3 SE 8πω Nρ ε 3 SE dω ω / d 1 N 8 N ( ) dω π ρε ρε d I dωdω c 3 6π Ng a δ L u p λγ N dω λ / Σ N 3 N g d dω n ρε π Effctiv noi pctra.(l.h.yu) Undulator radiation on axi (from L.W.potntial)

12 FLS006 DESY For th idal ca All - ar modulatd, th pontanou miion i full cohrnt: a rλ a δ ( ) N u u p CSE, l γσ ( klρ) ρ CSE CSE SE N l, For long bunch ( l b >S): ls CSE (4 π Nρ) ρ N CSE SE, numbr of - in th lippag ditanc

13 FLS006 DESY Cohrnt nhancmnt factor of CHG (cohrnt harmonic gnration) xiting thory: L.-W. potntial pontanou miion cohrnt nhancmnt factor cohrnt radiation FEL quation cohrnt radiation * a~ (prviou ) incohrnt pontanou radiation Cohrnt nhancmnt factor r λ a δ n L u p ( ) J n ( n ξ ) f r ( 4πNρ ) J n ( n ξ ) f r ρ γ a 16 πnλ rγρ / Σ 3 SE 8πω Nρ ε 3 SE R n N, J n ( n ξ ) f r N, : numbr of - in th lippag ditanc for radiator ction undulator of CHG. *JiaQika,; hy.rv.st.accl.bam, Vol.8, No.6(005)

14 FLS006 DESY Effctiv Start-up owr of SASE Whn thr xit an optical fild intraction with th -, - ditribution cannot b rgardd a indpndnt on th tim. λra uδ p iϕ a dϕ f(, τ Z, ϕ) τ γσ (1 β) + ϕ f 0 τ Z ϕ f f 0 + f 1, mono-nrgtic -: 1 z f δ S ζ + z Z δ τ Z + ζ giv pontanou miion 0 j0 0 ( ( j0 0 )) ( ) j 1 β j 1 β kkaδ f φ u u p τ iφ 0 i φ( τ τ') f1 R( a 0 d γ τ ') prturbing trm << f 0

15 FLS006 DESY ( k ρ) f a a a d d d a 3 τ τ' u p + τ ' φ τ" χ φ 0 0 i φ ( τ' τ") a 0 : input optical fild; a p : pontanou miion, (prviou) χ : avrag linar dnity of - Conidr th coating bam, th mono-nrgtic - abov q. can b olvd by Laplac tranform m ( a + µ a ( µ ))( µ + iφ ') i( a + µ a ( µ )) z a ( τ) R ( k ) ( ') ( ) ( )( ) µ z 3 µ 0 p m p m 3 uρ µ µ + iφ0 i kuρ m 1 µ m µ m µ l µ m µ k m l, k r a 1 i ik a ( ) ( λ δ µ + φ ) xp[ Z] xp[ ( µ + µ )( ζ + z )] H( ζ + z Z) u p u p j 0 j 0 γ Σ µ 1 β j 1 β µµ iφ i k u ρ 3 ( + 0 ') ( ) 0

16 FLS006 DESY th lading rol i th xponntial growth trm at th ronant nrgy φ 0 0, µ k ρ ( 3 + i ) 1 W obtain 1 a ( τ ) ( a 0 + a f ) 9 τ L g u a f ffctiv input powr of SASE

17 FLS006 DESY Effctiv input powr of SASE rλ a δ ( Z z ) a ( ) xp[ ] xp[ ( ρ 3 + i) k ζ ] H( ζ + z Z) u p 0 f j j γσ Lc j rλ a δ ( Z z ) ζ + u p 0 j ( ) xp( ){ xp( ) H( ζ j z0 Z) γσ Lc j Lc N + xp[ ( ρ 3 + ik ) ζ ( ρ 3 ik ) ζ ] H( ζ + z ZH ) ( ζ + z Z) } jl,( j l) j l j 0 l 0 0 a rλ a δ N N ( ) ( ) u p f Lc + γσ lb lbk Lc ( Lg / λu) λ, lippag ditanc pr L g

18 FLS006 DESY Th fir trm: ffctiv hot noi powr (incohrnt pontanou miion contribution ) a 4λ r n 3Σ γρ n ωρε 1 3 3N c, ρ 3 3 Comparing with prviou( ase 16 πnλrγρ / Σ, SE 8πω Nρ ε ) it i qual to th fraction of th pontanou undulator radiation in on L g frquncy domain approach: th ffctiv tart-up noi powr pctrum pontanou undulator radiation powr pctrum in 3 L g ~L g

19 FLS006 DESY Th cond trm: ffctiv upr-radianc powr (cohrnt pontanou miion contribution ) r 3 4ρ r n N 3 ρ N π, λ, c λ ( ) πl c N,λ and N,c : numbr of - in on λ and in on L c

20 FLS006 DESY SASE aturation timat Nar th aturation on can xpct th - ar approximatly full modulatd and maintaind in a ditanc αl g bfor aturation th radiation gnratd in thi ditanc α f ρ 3 Saturation powr (from prviou formula (4 π Nρ) ρ ) CSE α ρ α 3(1 ) 1.54ρ, α, th ditanc i th lat fild gain lngth 0.57ρ, α1, th ditanc i th lat powr gain lngth

21 FLS006 DESY Taking ρ, and only conidr hot noi ffctiv tart-up powr Saturation lngth L ln[7 N ] L (3.5 + ln[ IA ( ) λ ( nm)/ ρ]) L c, g g g. VISA FEL: λ 84nm, I50A, ρ L 0.3L g.075m (L g 10.cm for idal condition); L 3.63m (if L g 17.9cm for non-idal condition wr ud) agr with th xprimnt

22 FLS006 DESY With th tim domain approach, Summary Spontanou miion (incohrnt and cohrnt) for an arbitrary - pul profil. Th ffctiv tart-up powr of SASE Conit of: th hot noi trm, th incohrnt pontanou miion th uual pontanou undulator radiation in th on L g th upr radiant trm, th cohrnt pontanou miion. An analytical timation of aturation powr and lngth

23 FLS006 DESY