Motivation. TESLA SASE FELs. LCLS (seeded) SpontaneousSpectrum SASE FEL 1 SASE FEL 2. SPring8. U ndulator. (3 0m in vacuum )

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2 Motivation P.Gürtler,HASYLAB,Oct 02 Peak Brilliance [Phot./(sec m rad 2 m m 2 0.1% bandw.)] TESLA SASE FELs D ESY TTF FEL LCLS (seeded) SpontaneousSpectrum SASE FEL 1 D ESY TTF FEL 20 G ev SpontaneousSpectrum 10 G ev SASE FEL 2 TESLA spontaneous U ndulator SPring8 U ndulator (3 0m in vacuum ) ESRF U ndulator TTF FEL (I D 23) spontan BESSY II A PS U 49 U ndulator BESSY II (Typ A ) PETRA U 125 U ndulator A LS U PhotonEnergy [ev] Based on R.Brinkmann et al. (Ed.) Tesla FEL. Technical design report. Supplement.

3 Motivation P.Gürtler,HASYLAB,Oct 02 Peak Brilliance [Phot./(sec m rad 2 m m 2 0.1% bandw.)] TESLA SASE FELs D ESY TTF FEL LCLS (seeded) SpontaneousSpectrum SASE FEL 1 D ESY TTF FEL 20 G ev SpontaneousSpectrum 10 G ev SASE FEL 2 TESLA spontaneous U ndulator SPring8 U ndulator (3 0m in vacuum ) ESRF U ndulator TTF FEL (I D 23) spontan BESSY II A PS U 49 U ndulator BESSY II (Typ A ) PETRA U 125 U ndulator? A LS U PhotonEnergy [ev]

4 Crystalline Undulator y Trajectory of a channeling particle Radiation a d x z Λ u Periodically bent crystallographic plane A.V. Korol, A.V. Solov yov, W. Greiner, J. Phys. G 24, L45 (1998); Int. J. Mod. Phys. E 8, 49 (1999).

5 Crystalline Undulator P.Gürtler,HASYLAB,Oct 02 Peak Brilliance [Phot./(sec m rad 2 m m 2 0.1% bandw.)] TESLA SASE FELs D ESY TTF FEL LCLS (seeded) SpontaneousSpectrum SASE FEL 1 D ESY TTF FEL 20 G ev SpontaneousSpectrum 10 G ev SASE FEL 2 TESLA spontaneous U ndulator SPring8 U ndulator (3 0m in vacuum ) ESRF U ndulator TTF FEL (I D 23) spontan BESSY II A PS U 49 U ndulator BESSY II (Typ A ) PETRA U 125 U ndulator CU A LS U PhotonEnergy [ev] A.V. Korol, A.V. Solov yov, W. Greiner Topics in Heavy Ion Phys. (2005) A.V. Korol, A.P. Kostyuk, A.V. Solov yov, W. Greiner (2008), in preparation

6 Particles in an Undulator x or y z/! E 2 0 = E E E E ne 2 01 (1) Radiation Flux = c E 2 0 8π n (2)

7 Undulator vs Laser x or y z/! x or y

8 Particles in a Laser x or y z/! E 0 = E 01 + E 02 + E 03 + E 04 + ne 01 (3) Radiation Flux = c E 2 0 8π n2 (4)

9 Expected Brilliance P.Gürtler,HASYLAB,Oct 02 Peak Brilliance [Phot./(sec m rad 2 m m 2 0.1% bandw.)] TESLA SASE FELs D ESY TTF FEL LCLS (seeded) SpontaneousSpectrum SASE FEL 1 D ESY TTF FEL 20 G ev SpontaneousSpectrum 10 G ev SASE FEL 2 TESLA spontaneous U ndulator SPring8 U ndulator (3 0m in vacuum ) ESRF U ndulator TTF FEL (I D 23) spontan BESSY II A PS U 49 U ndulator BESSY II (Typ A ) PETRA U 125 U ndulator CUL CU A LS U PhotonEnergy [ev]

10 The question to be answered Will the beam preserve its modulation in a crystalline channel at sufficiently large penetration depth?

11 Channeling Oscillations y Trajectory of a channeling particle Radiation a d x z Λ u Periodically bent crystallographic plane

12 Beam Demodulation x y/a z/!

13 Demodulation within one Undulator Period Density of Channeling Particles (a.u.) z/! u

14 Demodulation within one Undulator Period Density of Channeling Particles (a.u.) z/! u

15 Demodulation within Dozens of Undulator Periods Density of Channeling Particles (a.u.) z/! u

16 Demodulation within Dozens of Undulator Periods Density of Channeling Particles (a.u.) z/! u

17 Quantitative Analysis is Necessary What is the characteristic length at which the beam gets demodulated? Is it possible to build a crystalline undulator with L dm /λ u 1?

18 Particle Distribution f (t, z; ξ, E y ) is the distribution function of channeling particles with respect to the angle x p ξ = p x p " and the transverse energy y z E y = p2 y 2E + U(y) U(#)/U(d/2) #/d The distribution depends on the penetration depth z and time t.

19 Kinetic Equation of Fokker-Planck Type 1 f c t + v z c Longitudinal Velocity: v z = c [ f z = D 0 E y D 0 = mc2 8 n e c ( 1 1γ2 1 ξ2 2 ( ) f E y ] f E y E ξ 2 dθ dσ dθ Θ2 ( 1 1 2γ 2 ξ2 2 E y 2E )( 1 E y 2E ) )

20 Kinetic Equation of Fokker-Planck Type 1 f c t + v z c Longitudinal Velocity: v z = c [ f z = D 0 E y D 0 = mc2 8 n e c ( 1 1γ2 1 ξ2 2 ( ) f E y ] f E y E ξ 2 dθ dσ dθ Θ2 ( 1 1 2γ 2 ξ2 2 E y 2E )( 1 E y 2E ) )

21 Kinetic Equation of Fokker-Planck Type 1 f c t + v z c Longitudinal Velocity: v z = c [ f z = D 0 E y D 0 = mc2 8 n e c ( 1 1γ2 1 ξ2 2 ( ) f E y ] f E y E ξ 2 dθ dσ dθ Θ2 ( 1 1 2γ 2 ξ2 2 E y 2E )( 1 E y 2E ) )

22 Excluding the Time Variable If the initial beam is periodically modulated it can be represented as a Fourier series with respect to the time: f (t, z; ξ, E y )= j= g j (z; ξ, E y ) exp(ijωt). where g j (z; ξ, E y )=g j (z; ξ, E y ) It is sufficient to consider only one harmonic. We substitute f (t, z; ξ, E y )=g(z; ξ, E y ) exp(iωt) i ω c g(z; ξ, E y )+ v [ ( ) z g c z = D g 0 E y + 1 E y E y E 2 ] g ξ 2

23 Separation of variables D 0 1 d 2 Ξ(ξ) E Ξ(ξ) dξ ( 2 ) D 0 d de(e y ) E y E(E y ) de y de y iω ξ2 2 = C ξ iω E y 2E = C y 1 dz(z) + iω Z(z) dz 2γ 2 = C z C z = C y + C ξ

24 The Solution g(z; ξ, E y ) = exp ( i ω ) c z Ξ n (ξ) =H n ( n=0 k=1 e iπ/8 4 ωe 2cD 0 ξ H n (...) are Hermite s polynomials. ( E k (E y )=exp 1+i 2 L νk (...) is Laguerre s function. c n,k Ξ n (ξ)e k (E y )Z n,k (z), ) ω cd 0 E E y exp ( 1+i 4 ) ( L νk (1 + i) ) ωe ξ 2. cd 0 ω cd 0 E E y )

25 The Solution g(z; ξ, E y ) = exp ( i ω ) c z n=0 k=1 c n,k Ξ n (ξ)e k (E y )Z n,k (z), [ ] Z n,k (z) = exp { zld κ α k (κ) + (2n + 1) j 0,1 [ ]} 1 iωz 2γ 2 + θ2 L β k(κ) + θl 2 (2n + 1) 2j 0,1 κ Dechanneling length: L d =4U max /(j 2 0,1 D 0) J 0 (j 0,1 ) = 0 Lidhard s angle: θ L = 2U max /E κ = π L d λ θ2 L

26 The Transcendental Equation for α k (κ) and β k (κ) κ = π L d λ θ2 L ( ) (1 + i) L νk j 0,1 κ =0. 2 α k (κ) = β k (κ) = κ (2R [(1 + i)ν k ] + 1) j 0,1 1 (2I [(1 + i)ν k ] + 1) 2j 0,1 κ

27 Asymptotic Behaviour g(z; ξ, E y ) = exp ( i ω ) c z n=0 k=1 c n,k Ξ n (ξ)e k (E y )Z n,k (z), [ ] Z n,k (z) = exp { zld κ α k (κ) + (2n + 1) j 0,1 [ ]} 1 iωz 2γ 2 + θ2 L β k(κ) + θl 2 (2n + 1) 2j 0,1 κ g(z; ξ, E y ) exp ( i ω ) c z c 0,1 Ξ 0 (ξ)e 1 (E y )Z 0,1 (z),

28 Asymptotic Particle Distribution vs. ξ and z

29 Asymptotic Particle Distribution vs. E y and z for κ = 10

30 Asymptotic Behaviour Demodulation Length: g(z; ξ, E y ) exp ( z/l dm iω/u z z) L dm = L d α 1 (κ) + κ/j 0,1 Phase velocity: u z = [ 1+ 1 ( )] 2γ θ2 L β k (κ) + 2j 0,1 κ κ = π L d λ θ2 L

31 Demodulation Length L dm /L d $

32 Parameter κ $ C(100) Diamond C(110) Diamond C(111) Diamond C(002) Graphite Si(100) Si(111) Ge(100) W(100) %h& (MeV)

33 Summary Beam demodulation is an important factor determining the physical limits on the applicability domain of the crystalline undulator based hard X ray and gamma ray laser. The beam evolution can be described by a partial differential equation of Fokker-Planck type which can be solved analytically for a parabolic interplanar potential. A channeling beam can preserve its modulation at sufficiently large penetration depths so that coherent radiation in the crystalline undulator is feasible. The bent crystal laser is most suitable for the application in the photon energy range of hundreds kev where the demodulation length not much smaller than the dechanneling length.

34 Aknowlegements I thank the organizers for this great conference in this beautiful place.

35 Aknowlegements Thank you for your attention. This work has been supported by the European Commission (the PECU project, Contract No (NEST)).

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38 Gamma Klystron L 1 L 2 L 3